CENGAGE / G TEWANI MATHS SOLUTIONS

CHAPTER SEQUENCES AND SERIES || ALGEBRA

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1

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

Write down the sequence whose nth term is and (ii)

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2

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

Find the sequence of he numbers defined by

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3

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

Write the first three terms of the sequence defined by .

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4

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

Consider the sequence defined by If then find the value of

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5

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

The Fibonacci sequence is defined by and Find

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6

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Introduction

A sequence of integers satisfies for . Suppose the sum of first 999 terms is 1003 and the sum of thefirst 1003 terms is ­99. Find the sum of the first 2002 terms.

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7 CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

2n /n [3 + ( − 1)n]/3n

an = , whe ∩ isodd , whe ∩ iseven1

n

1

n

a1 = 2, an+1 =2an + 3

an + 2

an = an2 + bn + ⋅ a1 = 1, a2 = 5, anda3 = 11, a10.

1 = a1 = a2 an = an−1 + an−2 ,n > 2. , f or n = 5.an+1

an

a1 + a2 + ...... + an an+2 = an+1 − an n ≥ 1

Show that the sequence 9, 12, 15, 18, ... is an A.P. Find its 16th term and the general term.

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8

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Show that the sequence is an A.P. Find the nth term.

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9

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum of terms of the sequence

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10

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

How many terms are there in the A.P. 3, 7, 11, ... 407?

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11

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., the find the value of

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12

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In a certain A.P., 5 times the 5th terms is equal to 8 times the 8th term. Then prove that its 13th term s 0.

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13

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the term of the series which is numerically the smallest.

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14

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

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15

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Given two A.P. and Then find the number of terms which are identical.

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16 CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Consider two A.P. s: Find the number of common term. Also find the last common term.

log a, log(ab), log(ab2), log(ab3),

n (an), wherean = 5 − 6n, n ∈ N .

a, b, c, d, e a − 4b + 6c − 4d + e.

25, 22 , 20 , 183

4

1

2

1

4

2, 5, 8, 11......T60 3, 5, 79, .........T50.

S1 : 2, 7, 12, 17, 500terms andS1 : 1, 8, 15, 22, 300terms

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17

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If pth, qth, and rth terms of an A.P. are respectively, then show that

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18

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then prove that are also in A.P.

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19

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If form an A.P., then prove that are also in A.P.

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20

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then prove that the following are also in A.P.

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21

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then prove that the following are also in A.P.

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22

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then prove that the following are also in A.P.

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23

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.

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a, b, c, (a − b)r + (b − c)p + (c − a)q = 0

(b + c − a)/a, (c + a − b)/b, (a + b − c)/c 1/a, 1/b, 1/c

a, b, c ∈ R + a + 1/(bc), b + 1/(1/ac), c + 1/(ab)

a, b, c a2(b + c), b2(c + a), c2(a + b)

a, b, c , ,1

√b + √c

1

√c + √a

1

√a + √b

a, b, c a( + ), b( + ), c( + )1

b

1

c

1

c

1

a

1

a

1

b

24

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Show terms of an A.P. is equal to twice the mth terms.

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25

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Divide 32 into four parts which are in A.P. such that the ratio of the product of extremes to the product of means is 7:15.

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26

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The digits of a positive integer, having three digits, are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than theoriginal number. Find the number.

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27

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of the series 2, 5, 8, 11, ... is 60100, then find the value of

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28

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In an A.P. if , then find the value of in terms of

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29

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Prove that the sum of number of terms of two different A.P. s can be same for only one value of

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(m + n)thand(m − n)th

n.

S1 = T1 + T2 + T3 + .... . + Tn(nodd).S2 = T2 + T4 + T6 + ......... + Tn−1 S1 /S2 n.

n n.

30

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In an A.P. of 99 terms, the sum of all the odd­numbered terms is 2550. Then find the sum of all the 99 terms of the A.P.

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31

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the degree of the expression

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32

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the number of terms in the series the sum of which is 300. Explain the answer.

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33

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum of all three­digit natural numbers, which are divisible by 7.

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34

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Prove that a sequence in an A.P., if the sum of its terms is of the form are constants.

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35

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sequence forms an A.P., then prove that

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(1 + x)(1 + x6)(1 + x11)....... . (1 + x101).

20, 19 , 18 ...1

3

2

3

n An2 + Bn, whereA,B

a1, a2, a3, an, . a12 − a22 + a32 + + a42 = (a12 − a2n2)n

2n − 1

36

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum of first 24 terms of the A.P. if it is inown that

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37

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the arithmetic progression whose common difference is nonzero the sum of first terms is equal to the sum of next terms. Then, find the ratio of thesum of the terms to the sum of next terms.

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38

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of terms of two arithmetic progressions are in the ratio Find the ratio of their 18th terms.

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39

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Insert five arithmetic means between 8 and 26. or Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

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40

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If eleven A.M. ‘s are inserted between 28 and 10, then find the number of integral A.M. ‘s.

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41

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Between 1 and 31 are inserted arithmetic means of that the ratio of the 7th and means is 5:9. Find the value of Or Between 1 and 31 , numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and numbers is 5:9. Find the value of

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a − 1, a2, a3, a1 + a5 + a10 + a15 + a20 + a24 = 225.

3n n2n 2n

n 5n + 4: 9n + 6.

m (m − 1)th m.

m (m − 1)thm.

42

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is A.M. between then find the value of .

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43

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Which term of the G.P.

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44

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The first terms of a G.P. is 1. The sum of the third and fifth terms is 90. Find the common ratio of the G.P.

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45

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If then show that are in G.P.

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46

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The fourth, seventh, and the last term of a G.P. are 10, 80, and 2560, respectively. Find the first term and the number of terms in G.P.

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47

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Three numbers are in G.P. If we double the middle term, we get an A.P. Then find the common ratio of the G.P.

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an + bn

an−1 + bn−1aandb, n

2, 1, 1/2, 1/4. . . . . . . is1/128?

= = (x ≠ 0),a + bx

a − bx

b + cx

b − cx

c + dx

c − dxa, b, c, andd

48

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are inA.P., show that the pth, qth, and rth terms of any G.P. are in G.P.

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49

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P. prove that are in G.P.

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50

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P. show that .

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51

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Three non­zero numbers are in A.P. Increasing by 1 or increasing by 2, the numbers are in G.P. Then find

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52

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of three numbers in GP. Is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

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53

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P. are in G.P and are in G.P. Prove that are in G.P.

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p, q, andr

a, b, c, d (an + bn), (bn + cn), (cn + dn)

a, b, candd (ab + bc + cd) = (a2 + b2 + c2)(b2 + c2 + d2)

a, b, andc a c b.

a, b, c b, c, d , ,1

c

1

d

1

ea, c, e

54

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the continued product o three numbers in a G.P. is 216 and the sum of their products in pairs is 156, find the numbers.

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55

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

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56

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Determine the number of terms in a G.P., if

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57

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let e the sum, the product, adn the sum of reciprocals of terms in a G.P. Prove that

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58

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum to terms of the sequence

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59

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Prove that the sum to terms of the series

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a1 = 3, an = 96, andSn = 189.

S P R n P 2Rn = Sn.

n (x + 1/x)2, (x2 + 1/x)

2, (x3 + 1/x)

2, ,

n 11 + 103 + 1005 + is(10/9)(10n − 1) + n2.

60

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum of the following series: terms. terms.

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61

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Prove that in a sequence of numbers 49,4489,444889,44448889 in which every number is made by inserting 48­48 in the middle of previous as indicated,each number is the square of an integer.

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62

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is a function satisfying for all such that and find the value of .

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63

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is a polomial in , then find possible value of

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64

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum of the following series:

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65

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If each term of an infinite G.P. is twice the sum of the terms following it, then find the common ratio of the G.P.

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5 + 55 + 555 + ... → n . 6 + . 66 + . 666 + ... → n

f f(x + y) = f(x) × f(y) x, y ∈ N f(1) = 3n

∑x=1

f(x) = 120, n

p(x) = (1 + x2 + x4 + + x2n−2)/(1 + x + x2 + + xn−1) x n.

(√2 + 1) + 1 + (√2 − 1) + ∞

66

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If then prove that

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67

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Prove that

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68

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of infinite number of terms in G.P. is 20 and the sum of their squares is 100. Then find the common ratio of G.P.

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69

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is the geometric mean of then prove that

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70

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Insert four G.M.’s between 2 and 486.

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71

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the product o three geometric means between 4 and 1/4.

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x = a + a/r2 + ∞, y = b − b/r + b/r2 − ∞, andz = c + c/r2 + c/r4 + ∞, xy/z = ab/ ⋅

61 / 2 × 61 / 4 × 61 / 8∞ = 6.

G xandy + =1

G2 − x2

1

G2 − y2

1

G2

72

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find two numbers whose arithmetic mean is 34 and the geometric mean is 16.

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73

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If A.M. and G.M. between two numbers is in the ratio then prove that the numbers are in the ratio

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74

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is the A.M. of and and the two geometric mean are and then prove that

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75

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The 8th and 14th term of a H.P. are 1/2 and 1/3, respectively. Find its 20th term. Also, find its general term.

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76

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the 20th term of a H.P. is 1 and the 30th term is ­1/17, then find its largest term.

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77

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P., then prove that and , are in A.P.

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m :n (m + √m2 − n2) :(m − √m2 − n2).

a b c G1 G2, G31 + G3

2 = 2ab ⋅

a, b, candd (b + c + d)/a, (c + d + a)/b, (d + a + b)/c (a + b + c)/d

78

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The mth term of a H.P is and the nth term is . Proves that its rth term is

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79

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If and are in G.P. then show that are in H.P.

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80

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If be in G.P. and in H.P. then find the value of .

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81

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If first three terms of the sequence are in geometric series and last three terms are in harmonic series, then find the values of

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82

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Insert four H.M.’s between 2/3 and 2/13.

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83

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is the harmonic mean between then find the value of

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n m mn/r.

a > 1, b > 1 c > 1 , , and1

1 + (log)ea

1

1 + (log)eb

1

1 + (log)ec

a, b, andc a + x, b + x, andc + x x(a, bandcaredist ∈ ctνmbers)

1/16, a, b,1

6aandb.

H PandQ H /P + H /Q.

84

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If nine arithmetic means and nine harmonic means are inserted between 2 and 3 alternatively, then prove that (where is any of theA.M.'s and the corresponding H.M.)

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85

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

The A.M. and H.M. between two numbers are 27 and 122, respectively, then find their G.M.

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86

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If the A.M. between two numbers exceeds their G.M. by 2 and the GM. Exceeds their H.M. by 8/5, find the numbers.

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87

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series terms.

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88

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of terms of the series

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89

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series upto terms.

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A + 6/H = 5 AH .

1 + 3x + 5x2 + 7x2 + → n

n 1 + + + + .4

5

7

5210

53

1 + 3x + 5x2 + 7x3 + .......... n

90

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to infinity of the series

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91

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series

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92

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series

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93

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series up to terms.

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94

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series

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95

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of first terms of the series is even is odd

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12 + 22x + 32x2 + ∞.

n 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + .

1 × n + 2(n − 1) + 3 × (n − 2) + + (n − 1) × 2 + n × 1.

+ + +13

1

13 + 23

1 + 3

13 + 23 + 33

1 + 3 + 5n

313 + 323 + + 503.

n 13 + 3 × 22 + 33 + 3 × 42 + 53 + 3 × 62 + when n n

96

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series

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97

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If then find the sum

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98

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If then find the sum of .

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99

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series

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100

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the following series to terms

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101

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series

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n 12 − 22 + 32 − 42 + 52 − 62 + . . . .

n

∑r=1

Tr = n(2n2 + 9n + 13),n

∑r=1

√Tr.

n

∑r=1

Tr = (3n − 1),n

∑r=1

1

Tr

n 3 + 15 + 35 + 63 +

n 5 + 7 + 13 + 31 + 85 +

n 1/(1 × 2) + 1/(2 × 3) + 1/(3 × 4) + + 1/n(n + 1).

102

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum

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103

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series

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104

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum to terms of the series that means find

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105

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum

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106

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum

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107

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum where n!= .

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n

∑r=1

.1

(ar + b)(ar + a + b)

n 3/(12 × 22) + 5/(22 × 32) + 7/(32 × 42) + .

n + + + ..........1

1 + 12 + 142

1 + 22 + 243

1 + 32 + 34tr =

r

r4 + r2 + 1

n

∑1

1 + + + .......... + .1

1 + 2

1

1 + 2 + 3

1

1 + 2 + 3 + ......... + n

n

∑r=1

1

r(r + a)(r + 2)(r + 3)

n

∑r=1

r

(r + 1)!1 × 2 × 3.... n

108

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum

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109

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum

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110

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series

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111

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series

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112

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of first 100 terms of the series whose general term is given by

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113

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the series terms.

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n

∑r=1

r(r + 1)(r + 2)(r + 3).

+ + + ...... +14

1 × 3

24

3 × 5

34

5 × 7

n4

(2n − 1)(2n + 1)

360

∑k=1

( )1

k√k + 1 + (k + 1)√k

+ + + + ∞1

32 + 1

1

42 + 2

1

52 + 3

1

62 + 4

ak = (k2 + 1)k !

+ × 2 + × 22 + × 23 + → n2

1 × 2

5

2 × 3

10

3 × 4

17

4 × 5

114

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

A sequence of numbers is defined as follows : and for each , then prove that

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115

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum of the products of the ten numbers taking two at a time.

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116

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum .

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117

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Prove that

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118

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Prove that .

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119

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Find the minimum value of .

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A∩ = 1, 2, 3 A1 =1

2n ≥ 2, An = ( )An−1

2n − 3

2nn

∑k=1

Ak < 1, n ≥ 1

±1, ± 2, ± 3, ± 4, and ± 5

n

∑r=0

^ (n + r)Cr

(ab + xy)(ax + by) > 4abxy(a, b, x, y > 0).

b2c2 + c2a2 + a2 + b2 > abc × (a + b + c)(a, b, c > 0)

4sin ^ (2x) + 4cos ^ (2x)

120

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If then prove that

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121

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are distinct positive real numbers such that then prove tha

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122

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive real number, then find the maximum value of

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123

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If Then find the minimum value of the expression

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124

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If then find the maximum value of

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125

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive, then prove that

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a + b + c = 1, > − 1 − 1 − 1 > 8.8

27abc

1

a

1

b

1

c

a, b, andc a + b + c = 1, > 8.(1 + a)(1 + b)(1 + c)

(1 − a)(1 − b)(1 − c)

(log)10(x3 + y3) − (log)10(x

2 + y2 − xy) ≤ 2, wherex, y xy.

(log)2(a + b) + (log)2(c + d) ≥ 4. a + b + c + ..

a+a2 + a3 + + an = 1∀a1 > 0, i = 1, 2, 3, , n, a12a2a3a4a5an.

a, b, c a/(b + c) + b/(c + a) + c/(a + b) ≥ 3/2.

126

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

In a triangle prove that

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127

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Prove that where is a positive integer.

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128

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Prove that

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129

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Find the least value of in an acute angled triangle.

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130

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If for then prove that

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131

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive values, find the greatest value of

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ABC a/(a + c) + b/(c + a) + c/(a + b) < 2

2n > 1 + n√2n−1, ∀n > 2 n

[n + 1/2]n > (n !).

secA + secB + secC

S + a1 + a2 + a3 + + an, a1 ∈ R+ i = 1 → n, + + + ≥ , ∀n ≥ 2S

S − a1

S

S − a2

S

S − an

n2

n − 1

yz + zx + xy = 12, wherex, y, z xyz.

132

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If such that then show that

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133

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If show that

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134

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

Prove that

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135

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

Prove that

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136

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If are positive and show that

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137

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that

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a, b, > 0 a3 + b3 = 2, a + b ≤ 2.

m > 1, n ∈ N 1m + 2m + 22m + 23m + + 2nm−m > n1−m(2n − 1)m

.

+ + > a + b + cb2 + c2

b + c

c2 + a2

c + a

a2 + b2

a + b

> + +a8 + b8 + c8

a3b3c31

a

1

b

1

c

a, b, andc a + b + c = 6, (a + 1/b)2 + (b + 1/c)2 + (c + 1/a)2 ≥ 75/4.

[ ]x+ y+ z

> xxyyzz > [ ]x+ y+ z

(x, y, z > 0)x2 + y2 + z2

x + y + z

x + y + z

3

138

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that

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139

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Find the greatest value of lie in the first quadrant on the line

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140

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Find the maximum value of lies between

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141

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If then prove that

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142

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If is a positive integer, then prove that

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143

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that ,where

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11 × 22 × 33 × × nn ≤ [(2n + 1)/3]n (n+1) /2

, n ∈ N .

x2y3, wherexandy 3x + 4y = 5.

(7 − x)4(2 + x)5whenx −2and7.

a1, a2, , an > 0, + + + + + > na1

a2

a2

a3

a3

a4

an−1

an

an

a1

a > bandn an − bn > n(ab)(n−1) /2

(a − b).

+ + < + +2

b + c

2

c + a

2

a + b

1

a

1

b

1

ca, b, c > 0.

144

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Find the minimum value of

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145

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If then show that

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146

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If are real numbers such that `0 Watch Free Video Solution on Doubtnut

147

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

In , prove that are acute angles.

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148

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

In prove that

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149

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If prove that `sum_(1lt=i Watch Free Video Solution on Doubtnut

2sinx2cosx.

a2 + b2 + c2 = x2 + y2 + z2 = 1, ax + by + cz < 1.

a, b, c

ΔABC tanA + tanB + tanC ≥ 3√3, whereA,B, C

ABC, cos ec + cos ec + cos ec ≥ 6.A

2

B

2

C

2

ai > 0(i = 1, 2, 3n),

150

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If is a positive integer then prove that

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151

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that the greatest value of is if

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152

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that

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153

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If the prove that `sqrt(C_1)+sqrt(C_2)+.......sqrt(C_n) lt sqrt(n(2^n­1)) ="">

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154

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If prove that

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155

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that

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n ≥ 1, > 1.3n

2n + n6( )n−12

xy c3 /√2ab. a2x4 + b2y4 = c6.

a4 + b4 + c4 > abc(a + b + c), wherea, b, c > 0.

Cr = ,n !

[r !(n − r)]

a + b = 1, a > 0, (a + )2

+ (b + )2

≥ .1

a

1

b

25

2

[ ]a+ b

> aabb > a+ b

.a2 + b2

a + b

a + b

2

156

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that

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157

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Prove that are distinct and

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158

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Given are positive rational numbers such that then prove that

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159

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

Find the greatest value of if are positive.

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160

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

For prove tht at most one term of the G.P. can be rational.

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161

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the terms of the A.P. are all in integers, then find the least composite value of

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apbq < ( )p+ q

.ap + bq

p + q

pxq− r + qxr−p + rxp− q > p + q + r, wherep, q, r x ≠ 1.

a, b, c a + b + c = 1, aabbcc + a^ ^ a + acbacb ≤ 1.

x2y3z4 x2 + y2 + z2 = 1, wherex, y, z

a, x, > 0 √a − x, √x, √a + x

√a − x, √x, √a + x wherea, x > 0, a.

162

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Along a road lie an odd number of stones placed at intervals of 10 metres. These stones have to be assembled around the middle stone. A person cancarry only one stone at a time. A man carried the job with one of the end stones by carrying them in succession. In carrying all the stones he covered adistance of 3 km. Find the number of stones.

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163

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the first and the nth terms of a G.P., are respectively, and if is hte product of the first terms prove that

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164

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let Find in terms of

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165

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If the sum terms of the series then find the value of

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166

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Find the sum terms.

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167

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are the sums of terms of A.P. whose first terms are and common differences are respectively.

Show that

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aandb, P n P 2 = (ab)n.

x = 1 + 3a + 6a2 + 10a3 + , |a| < 1. y = 1 + 4b + 10b2 + 20b3 + , |b| < 1. S + 1 + 3(ab) + 5(ab)2 + xandy.

n + 3( )2

+ 5( )3

+ is36,2n + 1

2n − 1

2n + 1

2n − 1

2n + 1

2n − 1n.

× + × ( )2

+ × ( )2

+ → n3

1 × 2

1

2

4

2 × 3

1

2

5

3 × 4

1

2

S1, S2, S3, Sm n m ' s 1, 2, 3, ,m 1, 3, 5, , (2m − 1)

S1 + S2, + Sm = (mn + 1)mn

2

168

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If be respectively the sum of n, 2n and 3n terms of a G.P., prove that

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169

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

In a sequence of terms, the first terms are n A.P. whose common difference is 2, and the last terms are in G.P. whose

common ratio is 0.5 if the middle terms of the A.P. and LG.P. are equal ,then the middle terms of the sequence is b. c. d.

none of these

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170

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

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171

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let be real numbers such that

then find the value of

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172

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If , x (log)_2 5 1­log_52 log_52` d. none of these

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173

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If three positive real numbers are in A.P. sich that , then the minimum value of is a. b. c. d.

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S1, S2andS3 S1(S3 − S2) = S1(S2 − S1)2

(4n + 1) (2n + 1) (2n + 1)n.2n + 1

22n − 1

n.2n + 1

2n − 1n.2n

∞

∑i=0

∞

∑j=0

∞

∑k=0

1

3i3i3k

a1, a2, .........an √a1 + √a2 − 1 + √a3 − 2 + +√an − (n − 1) = (a1 + a2 + ....... + an) − (n1

2

n − 3

4100

∑i=1

ai

(log)2(5 × 21−x + 1) (log)4(21−x + 1) and 1are ∈ A. P . , then equals b. c.

a, , b, c abc = 4 b 21 / 3 22 / 3 21 / 2 23 / 23

174

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The maximum sum of the series is b. c. d. none of these

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175

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The largest term common to the sequences terms and terms is b. c. d. none of these

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176

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of terms of an A.P. is the same as teh sum of its terms, then the sum of its terms is b. c. d.

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177

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If denotes the sum of terms of A.P., then b. c. d.

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178

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

About 150 workers were engaged to finish a piece of work in a certain number of days. Four workers stopped working on the second day, four moreworkers stopped their work on the third day and so on. It took 8 more days to finish the work. Then the number of days in which the work was completedis 29 days b. 24 days c. 25 days d. none of these

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179

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In an A.P. of which is the first term if the sum of the first terms is zero, then the sum of the next terms is a. b. c.

d. none of these

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20 + 19 + 18 +1

3

2

3310 300 0320

1, 11, 21, 31, → 100 31, 36, 41, 46, → 100 381 471 281

m n (m + n) mn −mn 1/mn 0

Sn n Sn+1 − 3Sn+2 + 3Sn+1 − Sn = 2S _ n sn+1 3Sn 0

a p qa(p + q)p

q + 1

a(p + q)p

p + 1

−a(p + q)q

p − 1

180

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If denotes the sum of first terms of an A.P. and , then the value of is 21 b. 15 c.16 d. 19

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181

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then is equal to b. c. d. none of these

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182

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The number of terms of an A.P. is even; the sum of the odd terms is 24, and of the even terms is 30, and the last term exceeds the first by 10/2 then thenumber of terms in the series is 8 b. 4 c. 6 d. 10

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183

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If and form two arithmetic progressions of the common difference, then are in A.P. if are in A.P. b. are in A.P. c.

are in G.P. d. none of these

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184

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Suppose that equals 50 b. 52 c. 54 d. none of these

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185

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of terms of an A.P. is where then the sum of the squares of these terms is b. c.

d. none of these

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Sn n = 31S3n − Sn−1

S2n − S2n−1n

a, b, andc a3 + c3 − 8b3 2abc 6abc 4abc

a, , c1

b, q,

1

p

1

ra, q, c p, b, r , ,

1

p

1

b

1

rp, b, r

F(n + 1) = (2 f or n = 1, 2, 3andF(1) = 2. Then.F(101)

F(n + 1)

2

n cn(n − 1) c ≠ 0, c2n(n + 1)2

c2n(n − 1)(2n − 1)2

3

n(n + 1)(2n + 1)2c2

3

186

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Consider an A.P. such that then the value of is ­8 b. 5 c. 7 d. 9

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187

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If be terms of an A.P. if equals 41/11 b. 7/2 c. 2/7 d. 11/41

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188

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If denote the sum of first terms of an A.P. whose first term is is independent of b. c. d.

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189

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is b. c. d.

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190

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are three successive terms of a G.P. with common ratio for which holds is given by `13orr<1` d. none ofthese

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191

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let denote the set of values of satisfying the equation . Then, b. c. d.

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a1, a2, a3, a3 + a5 + a8 = 11anda4 + a2 + = − 2, a1 + a + 6 + a + 7

a1, a2, a3, = , p ≠ q, thena1 + a2 + + ap

a1 + a2 + + aq

p2

q2a6

a21

Sn n aandSnx /Sx x, thenSp = p3 p2a pa2 a3

2 − √32 + √3 √3 − 2 3 + √2

a1, a2, a3(a1 > 0) r, a + 3 > 4a2 − 3a1

S ⊂ (0, π) x 81 + |cosx| + cos2 x + ∣ cos3x ∣ → ∞ = 43 S = π/3π/3, 2π/3 − π/3, 2π/3 π/3, 2π/3

192

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If then the sum of the series is b.

c. d.

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193

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If term of a G.P. is its term is , then its pth term is b. c. d. none of these

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194

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sides of a triangle are in G.P., and its largest angle is twice the smallest, then the common ratio satisfies the inequality `0 Watch Free Video Solution on Doubtnut

195

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The value of is b. c. d. none of these

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196

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If then is equal to b. c. d. none of these

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197

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P. nad , then b. c. d. none of these

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|a| < 1and|b| < 1, 1 + (1 + a)b + (1 + a + a2)b2 + (1 + a + a2 + a3)b3 + ...1

(1 − a)(1 − b)1

(1 − a)(1 − ab)

1

(1 − b)(1 − ab)

1

(1 − a)(1 − b)(1 − ab)

(p + q)th aand (p − q)th bwherea, b ∈ R+ √ a3

b√ b3

a√ab

r

0. 2log√5 + + +14

18

116 4 log 4 log 2

(1 + x)(1 + x2)(1 + x4)(1 + x128) =n

∑r=0

xr n 256 255 254

x, y, z ax = by = cz (log)ba = (log)ac (log)cb = (log)ac (log)ba = (log)cb

198

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

The geometric mean between ­9 and ­16 is b. c. d. none of these

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199

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If denotes the sum to infinity and the sum of terms of the series such that then the least value of is

b. c. d. 11

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200

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The first term of an infinite geometric series is The second term and the sum of the series are both positive integers. Then which of the following isnot the possible value of the second term b. c. d. none of these

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201

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The number of terms common between the series to terms and to terms is a. b. c. d.none of these

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202

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

After striking the floor, a certain ball rebounds (4/5)th of height from which it has fallen. Then the total distance that it travels before coming to rest, if it isgently dropped of a height of 120 m is b. c. d. none of these

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203

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747, then the common ratio of the G.P. is b. c. d. none of these

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12 −12 −13

S Sn n 1 + + + + ,1

2

1

4

1

8S − Sn < ,

1

1000n

8 9 10

21.12 14 18

1 + 2 + 4 + 8 + ....... 100 1 + 4 + 7 + 10 + ....... 100 6 4 5

1260m 600m 1080m

1/2 2/3 1/6

204

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P., then are in a. A.P. b. G.P. c. H.P. d. none of these

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205

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Consider the ten numbers If their sum is 18 and the sum of their reciprocals is 6, then the product of these ten numbers is a.b. c. d.

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206

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let then (a) (b) (c) (d)

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207

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let be the nth term of a G.P. of positive numbers. Let , such that , then the common ratio is b.

c. d.

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208

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of 20 terms of a series of which every even term is 2 times the term before it, every odd term is 3 times the term before it, the first tem being

unity is b. c. d. none of these

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209

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In a G.P. the first, third, and fifth terms may be considered as the first, fourth, and sixteenth terms of an A.P. Then the fourth term of the A.P., knowing thatits first term is 5, is b. c. d. 20

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a2 + b2, ab + bc, andb2 + c2 a, b, c

ar, ar2, ar3, .... . , ar10. 81243 343 324

a = 111....1(55digits), b = 1 + 102 + .... + 104, c = 1 + 105 + 1010 + 1015 + ... + 1050, a = b + c a = bc b = acc = ab

an

100

∑n=1

a2n = αand

100

∑n=1

a2n−1 = β α ≠ β α/β β/α

√α/β √β/α

( )(610 − 1)2

7( )(610 − 1)

3

7( )(610 − 1)

3

5

10 12 16

210

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the pth, qth, and rth terms of an A.P. are in G.P., then the common ratio of the G.P. is a. b. c. d.

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211

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the pth, qth, rth, and sth terms of an A.P. are in G.P., t hen are in a. A.P. b. G.P. c. H.P. d. none of these

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212

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P, then is equal to b. c. d.

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213

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are digits, then the rational number represented by ...is b. c. d.

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214

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of an infinite geometric series is 162 and the sum of its first terms is 160. If the inverse of its common ratio is an integer, then which of thefollowing is not a possible first term? b. c. d. none of these

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215

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let Then the number of real number of real values of for which the three unequal numbers are in G.P. is b. c. d. none of these

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pr

q2r

p

q + r

p + q

q − r

p − q

p − q, q − r, r − s

a, bc, d (b − c)2 + (c − a)2 + (d − b)2 (a − d)2 (ad)2 (a + d)2 (a/d)2

a, b, c ⊙ cababab cab/990 (99c + ba)/990 (99c + 10a + b)/99(99c + 10a + b)/990

n108 144 160

f(x) = 2x + 1. x f(x), f(2x), f(4x) 1 20

216

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

Concentric circles of radii are drawn. The interior of the smallest circle is colored red and the angular regions are colored alternatelygreen and red, so that no two adjacent regions are of the same color. Then, the total area of the green regions in sq. cm is equal to b. c.

d.

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217

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be a sequence of integers in G.P. in which . Then b. c. d. none of these

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218

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., where the distinct numbers are in G.P, then te common ratio of the G.P. is b. c. d.

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219

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If denotes the sum of the series the sum of the series in term of is

b. c. d.

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220

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are real and are in a. A.P. b. G.P. c. H.P. d. none of these

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221

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P., then are in a. A.P b. G.P. c. H.P. d. none of these

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1, 2, 3, . . . . , 100cm1000π 5050π

4950π 5151π

tn t4 : t6 = 1: 4andt2 + t5 = 216. t1is 12 14 16

x, 2y, 3z x, y, z 31

32

1

2

Sp 1 + rp + r2p + → ∞andsp 1 − r2pr3p + → ∞, |r| < 1, thenSp + sp S2p

2S2p 0 S2p1

2− S2p

1

2

x, y, z 4x2 + 9y2 + 16z2 − 6xy − 12yz − 8zx = 0, thenx, y, z

a1, a2, , an , , ,a1

a2 + a3 + + an

a2

a1 + a3 + + an

an

a1 + a2 + + an−1

222

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If harmonic means between 2 and 3, then a. 20 b.21 c. 40 d. 38

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223

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P. and are in a. A.P b. G.P. c. H.P. d. none of these

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224

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., the will be in a. A.P b. G.P. c. H.P. d. none of these

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225

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are the first three terms of a G.P., then the fourth term is a. 27 b. ­27 c. 13.5 d. ­13.5

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226

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of three numbers in G.P. is 14. If one is added to the first and second numbers and 1 is subtracted from the third, the new numbers are in ;A.P.The smallest of them is a. 2 b. 4 c. 6 d. 10

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227

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are in A.P. are in H.P., and are in G.P., then is equal to b. c. d.

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H1,H2, ,H20are20 + =H1 + 2

H1 − 2

H20 + 3

H20 − 3

a1, a2, a3, an f(k) = (n

∑r=1

ar) − ak, then , , , , ,a1

f(1)

a2

f(2)

a3

f(3)

an

f(n)

a, b, c , ,a

bc

1

c

2

b

x, 2x + 2, and3x + 3

a, b, andc p, q, andr ap, bq, andcr +p

r

r

p−

a

c

c

a+

a

c

c

a+

b

q

q

b−

b

q

q

b

228

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are in A.P. and are in G.P., then is b.1:3:5 c. d.

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229

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let satisfies the equation and Then sum of all possible values of S is a. b. c. d.

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230

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let If are the roots of quadratic equation is the roots of quadratic equation , then the

value of is the arithmetic mean of is b. c. d.

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231

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

such that are in A.P. and are in H.P., then b. c. d. none of these

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232

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If in a progression bears a constant ratio with , then the terms of the progression are in a. A.P b. G.P. c.H.P. d. none of these

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233

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P., then are in a. A.P b. G.P. c. H.P. d. none of these

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a, b, andc b − a, c − banda a : b : c 1: 2 : 3 2: 3 : 4 1: 2 : 4

a ∈ (0, 1] a2008 − 2a + 1 = 0 S = 1 + a + a2 + .... . + a2007 2010 20092008 2

α, β ∈ R. α, β2 x2 − px + 1 = 0andα2, β x2 − qx + 8 = 0

r ifr

8pandq,

83

283

83

8

83

4

a, b, cx ∈ R+ a, b, andc b, candd ab = cd ac = bd bc = ad

a1, a2, a3, .......... etc ., (ar − ar+1) ar × ar+1

a, b, andc a + b, 2b, andb + c

234

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are in A.P., are in G.P., and are in H.P. such that , then b. c. d.none of these

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235

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

Let Let deonote te arithmetic mean, geometric man, and harmonic mean of 25 and The least value of for which is a. 49 b. 81 c.169 d. 225

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236

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The 15th term of the series b. c. d. none of these

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237

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If are in G.P. and respectively, are the arithmetic means between , then the value of is b. c. d. none of

these

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238

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If are in A.P., and are respectively, A.M. and G.M. between are , respectively, the A.M. and G.M. between then b. c. d. none of these

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239

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P. with common difference , then the sum of the series is : a. b. c. d.

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a, x, andb a, y, andb a, z, b x = 9zanda > 0, b > 0 |y| = 3z x = 3|y| 2y = x + z

n ∈ N , n > 25. A,G,H n. nA,G,H ∈ 25, 26, n

2 + 1 + 1 + + is1

2

7

13

1

9

20

23

10

39

10

21

10

23

a, b, andc x, y, a, b, andb, c +a

x

c

y1 2 1/2

a, bandc pandp' aandbwhileq, q' bandc,p2 + q2 = p ' 2 + q ' 2 pq = p' q' p2 − q2 = p ' 2 − q ' 2

a1, a2, an d ≠ 0 sin d[sec a1sec a2 + (sec)2sec a3 + .... + secan−1(sec)n]cos ecan − cos eca cot an − cot a secan − seca tanan − tana

240

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of the series up to terms is b. c. d.

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241

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of 50 terms of the series is given by b. c. d. none of these

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242

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of 50 terms of the series is b. c. d.

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243

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If equals b. c. d.

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244

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

Coefficient of equal to b. c. d. none of these

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245

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

is equal to b. c. d. none of these

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a − (a + d) + (a + 2d) − (a + 3d) + (2n + 1) −nd a + 2nd a + nd 2nd

1 + 2(1 + ) + 3(1 + )2

+1

50

1

502500 2550 2450

+ + +3

125

12 + 227

12 + 22 + 32100

17

150

17

200

51

50

17

+ + + → ∞ = , then + + +1

121

221

32π2

6

1

121

321

52π2 /8 π2 /12 π2 /3 π2 /2

x18 ∈ (1 + x + 2x2 + 3x3 + + 18x18)2

995 1005 1235

( lim )n−→∞

n

∑r=1

r

1 × 3 × 5 × 7 × 9 × × (2r + 1)

1

3

3

2

1

2

246

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

The greatest integer by which is divisible is a. composite number b. odd number c. divisible by 3 d. none of these

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247

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If is equal to b. c. d.

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248

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Value of is equal to a. b. c. d. none of these

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249

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are in H.P. and are in G.P., then b. c. d. none of these

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250

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The value of is equal to b. c. d.

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251

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If b. c. d.

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1 +30

∑r=1

r × r !

n

∑r=1

r4 = I(n), thenn

∑r=1

(2r − 1)4

I(2n) − I(n) I(2n) − 16I(n) I(2n) − 8I(n) I(2n) − 4I(n)

(1 + )(1 + )(1 + )(1 + )....... . ∞1

3

1

321

341

383

6

5

3

2

x1, x2, x20 x1, 2, x20

19

∑r=1

xrxr+1 = 76 80 84

n

∑r=0

(a + r + ar)( − a)r ( − 1)n[(n + 1)an+1 − a] ( − 1)n(n + 1)an+1 ( − 1)n(n + 2)an+1

2( − 1)n

nan

2

bi = 1 − ai, na =n

∑i=1

ai, nb =n

∑i=1

bi, thenn

∑i=1

ai, bi +n

∑i=1

(ai − a)2= ab nab (n + 1)ab nab

252

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of series to infinite terms, if is b. c. d.

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253

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are in A.P., then is equal to b. c.

d. none of these

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254

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of up to 100 terms, where is b. c. d.

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255

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let . Then is equal to b. c. d. none of these

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256

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If , then the value of is b. c. d.

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257

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum to infinity of the series is 9/4, then value of is (a) b. c. d. none of these

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+ + +x

1 − x2

x2

1 − x4

x4

1 − x8|x| < 1,

x

1 − x

1

1 − x

1 + x

1 − x1

a1, a2, a3, , a2n+1 + + +a2n+1 − a1

a2n+1 + a1

a2n − a2

a2n + a2

an+2 − an

an+2 + an×

n(n + 1)

2

a2 − a1

an+1

n(n + 1)

2(n + 1)(a2 − a1)

i − 2 − 3i + 4 i = √−1 50(1 − i) 25i 25(1 + i) 100(1 − i)

S = + + + up → ∞4

19

44

192444

193s 40/9 38/81 36/171

Hn = 1 + + + .1

2

1

nSn = 1 + + + +

3

2

5

3

99

50H50 + 50 100 − H50 49 + H50 H50 + 100

1 + 2r + 3r2 + 4r3 + r 1/2 1/3 1/4

258

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum o f series is b. c. d.

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259

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If A.M., G.M., and H.M. of the first and last terms of the series of are the terms of the series itself, then the value of `ni s(100 Watch Free Video Solution on Doubtnut

260

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum terms is b. c. d. none of these

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261

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

In a sequence of terms, the first terms are n A.P. whose common difference is 2, and the last terms are in G.P. whose

common ratio is 0.5 if the middle terms of the A.P. and LG.P. are equal ,then the middle terms of the sequence is a. b. c. d.

none of these

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262

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

The coefficient of in the product b. c. d. none of these

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263

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of 20 terms of the series whose rth term s given by k is b. c. d. none of these

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1 + + + + ∞4

5

7

5210

537/16 5/16 104/64 35/16

100, 101, 102, ...n − 1, n

1 + 3 + 7 + 15 + 31 + ... → 100 2100 − 102b 299 − 101 2101 − 102

(4n + 1) (2n + 1) (2n + 1)n.2n + 1

22n − 1

n.2n + 1

2n − 1n.2n

x49 (x − 1)(x − 3)(x + 99)is −992 1 −2500

T (n) = ( − 1)n n

2 + n + 1

n !

20

19!− 1

21

20!

21

20!

264

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Consider the sequence 1,2,2,4,4,4,8,8,8,8,8,8,8,8,... Then 1025th terms will be b. c. d.

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265

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If for then b. c. d.

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266

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The sum of to is b. c. d. none of these

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267

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The value of then the value of equals a. b. c. d.

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268

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If equals b. c. d. 2001

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269

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If denotes the nth term of the series b. c. d.

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29 211 210 212

tn =1

4(n + 2)(n + 3)n = 1, 2, 3, , + + + + =

1

t1

1

t2

1

t3

1

t2003

4006

3006

3006

3007

4006

3008

4006

3009

0. 2 + 0004 + 0. 00006 + 0. 0000008 + ... ∞200

891

2000

9801

1000

9801

n

∑i=1

i

∑j=1

j

∑k=1

1 = 220, n 11 12 10 9

12 + 22 + 32 + + 20032 = (2003)(4007)(334)and(1)(2003) + (2)(2002) + (3)(2001) + + (2003)(1) = (2003)(334)(x), thenx2005 2004 2003

tn 2 + 3 + 6 + 11 + 18 + ...thent50 492 − 1 492 502 + 1 492 + 2

270

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The positive integer for which is a. b. c. d.

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271

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If , then value of is b. c. d.

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272

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

The coefficient of in the polynomial is b. c. d. none of these

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273

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If is equal to b. c. d. none of these

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274

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If , then is equal to b. c. d. none of these

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275

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If is divisible by are in a. A.P. b. G.P. c. H.P. d. none of these

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n 2 × 22 + 3 × 23 + 4 × 24 + ...... + n × 2n = 2n+10 510 511 512 513

1 − + − + − + =1

3

1

5

1

7

1

9

1

11

π

4+ + +

1

1 × 3

1

5 × 7

1

9 × 11π/8 π/6 π/4 π/36

x19 (x − 1)(x − 2)(x − 22)(x − 219) 220 − 219 1 − 220 220

bn+1 = f or n ≥ 1andb1 = b3, then2001

∑r=1

br20011

1 − bn2001 −2001 0

(12 − t1) + (22 − t2) + + (n2 − tn) + =n(n2 − 1)

3tn n2 2n n2 − 2n

ax3 + bx2 + cx + d ax2 + c, thena, b, c, d

276

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If where each set of parentheses contains the sum of consecutive odd integers asshown, the smallest possible value of is b. c. d.

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277

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

In a geometric series, the first term is and common ratio is If denotes the sum of the terms and

equals b. c. d.

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278

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The line meets X­axis at A and Y­axis at B,P is the mid­point of is the foot ofperpendicular from P to , is that of , from , is that of from , is that of , from is that of , from OA and so on. If denotes the nth foot of the perpendicular on OA,

then find .

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279

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

If , then the value of a b. c. d.

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280

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

is a right­angled triangle in which and If points is divided in equal parts and are line segments parallel to are on then the sum of the lengths of is

b. c. d. none of these

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281

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let be the rth term and sum up to rth term of a series, respectively. If for an odd number ( being

even)is b. c. d.

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(1 + 3 + 5 + + p) + (1 + 3 + 5 + + q) = (1 + 3 + 5 + + r)p + q + r(wherep > 6) 12 21 45 54

a r. Sn Un =n

∑n=1

Sn, thenrSn + (1 + = − r)Un

0 n na nar

x + y = 1 AB, P1 OA,M1 P1OP ; P2 M1 OA,M2 P2 OP ; P3 M2 Pn

OPn

(1 − p)(1 + 3x + 9x2 + 27x3 + 81x4 + 243x5) = 1 − p6, p ≠ 1 isp

x

1

33

1

22

ABC ∠B = 900 BC = a. n L1, L2, , LnonAB n + 1L1M1, L2M2, , LnMn BCandM1,M2, ,Mn AC, L1M1, L2M2, , LnMn

a(n + 1)

2

a(n − 1)

2

an

2

TrandSr n, Sn = nandTn = , thenTmTn − 1

n2m

2

1 + m2

2m2

1 + m2

(m + 1)2

2 + (m + 1)22(m + 1)2

1 + (m + 1)2

282

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

ABCD is a square of length a, , a > 1. Let be points on BC such that and be

points on CD such that . Then is equal to :

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283

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P. and are in H.P., then b. c. d.

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284

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are distinct prime numbers, then may be in A.P. but not in G.P. may be in G.P. but not in A.P. can neitherbe in A.P. nor in G.P. none of these

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285

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

For the series, +... 7th term is 16 7th term is 18

Sum of first 10 terms is Sum of first 10 terms is

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286

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If sum of an infinite G.P. then value of is a. 3 b. 3/2 c. 3 d. 9/2

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287

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If then are in A.P. is an integer

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a ∈ N L1, L2, L3... BL1 = L1L2 = L2L3 = .... 1 M1,M2,M3, ....

CM1 = M1M2 = M2M3 = ... = 1a−1

∑n=1

((ALn)2 + (LnMn)

2)

x, y, andz x + 3 + , y + 3, andz + 3 y = 2 y = 3 y = 1 y = 0

x, y, andz x, y, andz x, y, andz x, y, andz

S = 1 + (1 + 2)2+ (1 + 2 + 3)

2+ (1 + 2 + 3 + 4)

21

(1 + 3)

1

(1 + 3 + 5)

1

(1 + 3 + 5 + 7)505

4

45

4

p, 1, 1/p, 1/p2, is9/2 p

n

∑r=1

r(r + 1)(2r + 3) = an4 + bn3 + cn2 + dn + e, a − b = d − c e = 0 a, b − 2/3, c − 1 (b + d)/a

288

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The terms of an infinitely decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth terms is , then b. c. d. none of these

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289

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The consecutive digits of a three digit number are in G.P. If middle digit is increased by 2, then they form an A.P. If 792 is subtracted from this, then we getthe number constituting of same three digits but in reverse order. Then number is divisible by a. 7 b. 49 c. 19 d. noneof these

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290

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If b. c. d.

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291

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Given that are in A.P. and are in H.P. Then G.M. of is 3 One

possible value of is 11 A.M. of is 6 Greatest value of is 8

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292

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be in G.P. such that Then common ratio of G.P. can be b. c. d.

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293

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

terms is equal to b. c. less than d. less than

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32/81 r = 1/3 r = 2√2/3 S∞ = 6

Sn = 12 − 22 + 32 − 42 + 52 − 62 + , then S40 = − 820 S2n > S2n+2 S51 = 1326 S2n+1 > S2n−1

x + y + z = 15whena, x, y, z, b + + + = whena, x, y, z, b1

x

1

y

1

z

5

3aandb

a + 2b aandb a − b

a1, a2, a3, , an 3a1 + 7a2 + 3a3 − 4a5 = 0. 23

2

5

2−

1

2

+ + + n1

√2 + √5

1

√5 + √8

1

√8 + √11

√3n + 2 − √2

3

n

√2 + 3n + √2n √

n

3

294

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P., then th value of is b. c. d.

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295

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If is a polynomial in can be b. c. d.

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296

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

For an increasing A.P. and then which of the following is/are true? a. b. c. d.

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297

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If , the value of the positive integer for which divides is/are b. c. d.

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298

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then which of the following is/are true? pth, qth, and rth terms of A.P. are in A.P. pth, qth, rth terms of G.P. are in G.P. pth, qth, rthterms of H.P., are in H.P. none of these

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299

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If , then least value of greatest value of least value of greatest value of does not exists

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a, b, andc(ac + ab − bc)(ab + bc − ac)

(abc)2

(a + c)(3a − c)

4a2c2−

2

bc

1

b2−

2

bc

1

a2

(a − c)(3a + c)

4a2c2

p(x) =1 + x2 + x4 + + x

1 + x + x2 + + xn−1 ^ (2n − 2)x, the ∩ 5 10 20 17

a1, a2, an if a1 = a2 + a3 + a5 = − 12 a1a3a5 = 80, a1 = − 10a2 = − 1 a3 = − 4 a5 = + 2

n > 1 m nm + 1 a = 1 + n + n2 +..+ n63 8 16 32 64

p, q, andr

1 + 2x + 3x2 + 4x3 + ∞ ≥ 4 ξs1/2 ξs4

3ξs2/3 x

300

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let Then, a. b. c. d.

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301

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of terms of an A.P. is given by are independent of (a) (b) common difference of A.P.must be (c) common difference of A.P. must be (d) first term of A.P. is

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302

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P. b. are in A.P. c. are in G.P. d.

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303

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The next term of the G.P. is b. c. d.

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304

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let be squares such that for each the length of a side of equals the length of a diagonal of If the length of a side of then for which of the following value of is the area of less than 1 sq. cm? a. 5 b. 7 c. 9 d. 10

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305

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If then A. are in H.P. B. are in A.P. C. D.

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E = + + +1

121

221

32E < 3 E > 3/2 E > 2 E < 2

n Sn = a + bn + cn2, wherea, b, c n, then a = 02b 2c b + c

x2 + 9y2 + 25z2 = xyz( + + ), then15

x

5

y

3

zx, y, andz , ,

1

x

1

y

1

zx, y, z + = =

1

a

1

d

1

b

1

c

x, x2 + 2, andx3 + 10729

166 0 54

S1, S2, n ≥ 1, Sn Sn+1.S1is10cm, n Sn

+ = + ,1

b − a

1

b − c

1

a

1

ca, b, andc a, b, andc b = a + c 3a = b + c

306

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P. and respectively, be arithmetic means between b. c. d.

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307

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Consider a sequence for all terms of the sequence being distinct. Given that are positive

integers and then the possible value can be a. 162 b. 64 c. 32 d. 2

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308

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Which of the following can be terms (not necessarily consecutive) of any A.P.? a. 1,6,19 b. c. d.

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309

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The numbers 1, 4, 16 can be three terms (not necessarily consecutive) of no A.P. only on G.P. infinite number o A.P.’s infinite number of G.P.’s

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310

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Each question has four choices a,b,c and d out of which only one is correct. Each question contains Statement 1 and Statement 2. Make your answer as:If both the statements are true and Statement 2 is the correct explanation of statement 1. If both the statements are True but Statement 2 is not the

correct explanation of Statement 1. If Statement 1 is True and Statement 2 is False. If Statement 1 is False and Statement 2 is True. Statement 1:

is a root of Statement 2: For any

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311

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

if a G.P (p+q)th term = m and (p­q) th term = n , then find its p th term

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a, b, andc xandy, a, b, andb, c, then + = 2a

x

c

y+ =

a

x

c

y

c

a+ =

1

x

1

y

2

b

+ =1

x

1y

2

ac

anwitha1 = 2andan =an − 12

an−2n ≥ 3, a1anda5

a5 ≤ 162 (s)ofa5

√2, √50, √98 log 2, log 16, log 128 √2, √3, √7

sinπ

188x3 − 6x + 1 = 0 θ ∈ R, sin 3θ = 3 sin θ − 4 sin3 θ

312

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in H.P. b. are in A.P. c. are in G.P. d.

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313

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the roots of are in harmonic oprogresion, then eqauls _________.

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314

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The difference between the sum of the first terms of the series and the sum of the first terms of The value of is __________.

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315

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

The value of the is equal to_________.

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316

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let are in G.P. with Then the value of equals____________.

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317

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let , then equals ___________.

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x2 + 9y2 + 25z2 = xyz( + + ), then15

2

5

y

3

zx, y, andz , ,

1

x

1

y

1

zx, y, z + = =

1

a

1

d

1

b

1

c

10x3 − nx2 − 54x − 27 = 0 n

k 13 + 23 + 33 + + n3 k 1 + 2 + 3 + + nis1980.k

∞

∑n=0

2n + 3

3n

a1, a2, a3, , a101 a101 = 25and201

∑i=1

a1 = 625.201

∑i=1

1

a1

S =9999

∑n=1

1

(√n + √n + 1)(n4 + n + 14)S

318

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The 5th and 8th terms of a geometric sequence of real numbers are 7! And 8! Respectively. If the sum to first tems of the G.P. is 2205, then equals_______.

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319

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be four distinct real numbers in A.P. Then half of the smallest positive valueof satisfying is __________.

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320

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The number of positive integral ordered pairs of such that are in harmonic progression is _________.

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321

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let let be in A.P. and are in G.P., then the value of is _______.

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322

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The coefficient of the quadratic equation are consecutive terms of a positively valued, increasing arithmetic sequence.Then the least integral value of such that the equation has real solutions is __________.

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323

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be two infinite series of positive numbers with the same first term. The sum of the firstseries is and the sum of the second series Then the value of is ________.

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n n

a, b, c, d k

2(a − b) + k(b − c)2 + (c − a)3 = 2(a − d) + (b − d)2 + (c − d)3

(a, b) 6, a, b

a, b > 0, 5a − b, 2a + b, a + 2b (b + 1)2, ab + 1, (a − 1)2 (a−1 + b−1)

ax2 + (a + d)x + (a + 2d) = 0d/a

a + ar1 + ar12 + + ∞anda + ar2 + ar22 + + ∞r1 r2. (r1 + r2)

324

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If he equation has three real roots in G.P., then has the value equal to _____.

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325

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be a geometric sequence. Define as the product of the first terms. Then the value of is _________.

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326

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The terms from an arithmetic sequence whose sum s 18. The terms in that order, form a geometric sequence. Then theabsolute value of the sum of all possible common difference of the A.P. is ________.

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327

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Given are in A.P. are in G.P. and are in H.P. If then the sum of all possible value of is ________.

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328

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let the sum of first three terms of G.P. with real terms be 13/12 and their product is ­1. If the absolute value of the sum of their infinite terms is then thevalue of is ______.

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329

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let denote sum of the series Then the value of is __________.

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x3 + ax2 + bx + 216 = 0 b/a

an = 16, 4, 1, Pn n

∞

∑n=1

Pn1

4

1n

a1, a2, a3 a1 + 1, a2, a3, + 2,

a, b, c b, c, d c, d, e a = 2ad ≠ = 18, c

S,7S

S + + + + ∞3

234

24.3

5

26.3

6

27.5S −1

330

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The first term of an arithmetic progression is and the sum of the first nine terms equal to . The first and the ninth term of a geometric progressioncoincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

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331

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

The harmonic mean of two numbers is 4. Their arithmetic mean and the geometric mean satisfy the relation Find two numbers.

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332

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The interior angles of a polygon are in arithmetic progression. The smallest angle is and the common difference is Find the number of sides ofthe polygon

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333

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., where for all , show that

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334

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

How many geometric progressions are possible containing 27, 8 and 12 as three of its/their terms

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335

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

9. Find three numbers a,b,c between 2 & 18 such that; (G) their sum is 25 (ii) the numbers 2,a,b are consecutive terms of an AP & (ii) the numbers b,c, 18are consecutive terms of a G.P.

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1 369

A G 2A + G2 = 27.

120 ∘ 5 ∘

a1, a2, a3, , an ai > 0 i + + + = .1

√a1 + √a2

1

√a1 + √a3

1

√an−1 + √an

n − 1

√a1 + √an

336

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The sum of the first three terms of a strictly increasing G.P. is and sum of their squares is

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337

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in arithmetic progression, determine the value of

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338

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

If p is the first of the n arithmetic means between two numbers and q be the first on n harmonic means between the same numbers. Then, show that q

does not lie between p and

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339

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are the sums of infinite geometric series whose first terms are and whose common ratio

respectively, then find the value of .

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340

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

The real numbers satisfying the equation ar ein A.P. Find the intervals in which lie.

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341

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

Let be real numbers in If satisfy the system of equations then

show that the roots of the equation and 20x^2+10(a­d)^2 x­9=0` are

reciprocals of each other.

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αs s2

(log)32, (log)3(2x − 5)and(log)3(2

x − )7

2x.

( )2

p.n + 1

n − 1

S1, S2, S3, .........Sn, ....... . 1, 2, 3............n, .............

, , , ....... . , , ....1

2

1

3

1

4

1

n + 1

2n−1

∑r=1

S21

x1, x2, x3 x3 − x2 + bx + γ = 0 βandγ

a, b, c, d G. P . u, v, w u + 2y + 3w = 6, 4u + 5v + 6w = 12 and 6u + 9v = 4

( + )x2 + [(b − c)2 + (c − a)2 + (d − b)2]x + u + v + w = 01

u

1

v

+

w

342

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

The fourth power of common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Provethat the resulting sum is the square of an integer.

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343

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

.Let be positive real numbers in geometric progression. For each n, let , be respectively the arithmetic mean, geometric mean& harmonic mean of . Find an expression ,for the geometric mean of in terms of

.

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344

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be positive real numbers. If be are in arithmetic progression are in geometric progression, and are in

harmonic progression, show that

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345

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P. and are in H.P., then prove that either form a G.P.

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346

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If and , then find the minimum natural number n, such that

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347

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If are pth, qth, and rth terms, respectively, of an A.P. nd also of a G.P., then is equal to a. b. c. d. none of these

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a1, a2, ............ AnGn,Hn

a1, a2.......... an G1,G2, ....... .Gn

A1,A2, ....... . ,An,H1,H2, ....... . ,Hn

a, b aA1,A2, b a,G1,G2, b a,H1,H2, b

=G1G2

H1H2

A1 + A2

H1 + H2

a, b, c a2, b2, c2 a = b = c or a, b, −c

2

an = − ( )2

+ ( )3

+ ...( − 1)n−1( )n

3

4

3

4

3

4

3

4bn = 1 − an bn > an

x, y, andz xy− zyz−xzx− y xyz 0 1

348

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

The third term of a geometric progression is 4. The production of the first five terms is b. c. d. none of these

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349

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

In triangle ABC medians AD and CE are drawn, if AD=5, and , then the area of triangle ABC is equal to a. b. c.

d.

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350

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P., then the equations have a common root if are in a. A.P. b. G.P. c. H.P.

d. none of these

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351

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Sum of the first terms of the series is equal to b. c. d.

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352

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Find the sum b. c. d. none of these

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353

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in A.P., then (a) are in A.P. (b) are in A.P. (c) are in G.P. d. (d) are in H.P.

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43 45 44

∠DAC =π

8∠ACE =

π

4

25

8

25

3

25

1810

3

a, b, andc ax2 + 2bx + c = 0anddx2 + 2ex + f = 0 , ,d

c

e

b

f

c

n + + + +1

2

3

4

7

8

15

162n − n − 1 1 − 2−n n + 2−n − 1 2n + 1

(x + 2)n−1 + (x + 2)n−2(x + 1)+ (x + 2)n−3(x + 1)2 + + (x + 1)n−1 (x + 2)n−2 − (x + 1)n (x + 2)n−2 − (x + 1)n−1

(x + 2)n− (x + 1)

n

ln(a + c), ln(a − c)andln(a − 2b + c) a, b, c a2, b2, c2, a, b, c a, b, c

354

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be in A.P. and be in H.P. If is b. c. d.

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355

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Properties Of A.M. And G.M. And H.M.

The harmonic mean of the roots of the equation is b. c. d.

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356

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let the positive numbers be in the A.P. Then are a. not in A.P. /G.P./H.P. b. in A.P. c. in G.P. d. in H.P.

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357

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Consider an infinite geometric series with first term and common ratio If its sum is 4 and the second term is 3/4, then b.

c. d.

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358

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be the roots of be the root of If are in G.P., then the integral values of , respectively, are b. c. d.

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359

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If the sum of the first terms of the A.P. 2, 5, 8, ..., is equal to the sum of the first terms of A.P. 57, 59, 61, ..., then equals b. c. d.

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a1, a2, , a10 h1, h2, h10 a1 = h1 = 2anda10 = h10 = 3, thena4h7 2 3 5 6

(5 + √2)x2 − (4 + √5)x + 8 + 2√5 = 0 2 4 6 8

a, b, cadnd abc, abd, acd, andbcd

a r. a = , r =4

7

3

7

a = 2, r =3

8a = , r =

3

2

1

2a = 3, r =

1

4

αandβ x2 − x + p = 0andγandδ x2 − 4x + q = 0. α, β, andγ, δ pandq−2, − 32 −2, 3 −6, 3 −6, − 32

2n n n 10 12 11 13

360

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

An infinite G.P. has first term as and sum 5, then

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361

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

in the quadratic , and , , , are in G.P , where are the roots of , then

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362

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be in harmonic progression with The least positive integer for which a. b. c. d.

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363

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If first and terms of an AP, GP. and HP. are equal and their nth terms are a, b, c respectively, then (a) a=b=c (b)a+c=b (c) a>b>c and (d) none of these

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364

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

If `x=sum_(n=0)^oocos^(2n)theta,y=sum_(n=0)^oosin^(2n)varphi,z=sum_(n=0)^oocos^(2n)thetasin^(2n)varphi,w h e r e0 Watch Free Video Solution on Doubtnut

365

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Q. Let n be an add integer if sin ntheta­sum_(r=0)^n(b_r)sin^rtheta, for every value of theta then ­­­

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a

ax2 + bx + c = 0 D = b2 − 4ac α + β α2 + β2 α3 + β3 α, β ax2 + bx + c

a1, a2, a3, ... a1 = 5anda20 = 25. n an < 0 22 23 24 25

(2n − 1)thac − b2 = 0

366

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be the rth term of an A.P., for If for some positive integers we have equals a. b.

c. d.

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367

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

If are in G.P., then are in b. c. d. none of these

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368

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

For a positive integer let Then b. c. d.

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369

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let Then can take value (s) b. c. d.

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370

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let denotes thesum of the infinite geometric series whose first term s and the common ratio is , then the value of

is _______.

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371

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Le be real numbers satisfying for If

then the value of is equals to _______.

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Tr r = 1, 2, 3, .... . m, n, Tm = andTn = , thenTmn1

n

1

m

1

mn

+1

m

1

n1 0

x > 1, y > 1, andz > 1 , and1

1 + lnx

1

1 + lny

1

1 + lnzA

.P . H

.P . G

.P .

n a(n) = 1 + + + + .1

2

1

3

1

4

1

(2n) − 1a(100) ≤ 100 a(100)

.> 100 a(200) ≤ 100 a(200) ≤ 100

Sn =4n

∑k=1

( − 1) k2.k(k + 1)

2Sn 1056 1088 1120 1332

Sk, k = 1, 2, , 100,k − 1

k !

1

k

+100

∑k=1

(k2 − 3k + 1)Sk

1002

100!

a1, a2, a3, , a11 a1 = 15, 27 − 2a2 > 0andak = 2ak−1 − ak−2 k = 3, 4, , 11.

= 90,a12 + a22 + ... + a112

11

a1 + a2 + + a11

11

372

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Let be an arithmetic progression with For any integer with let If

does not depend on then is__________.

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373

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

A pack contains cards numbered from 1 to . Two consecutive numbered cards are removed from the pack and the sum of the numbers on theremaining cards is 1224. If the smaller of het numbers on the removed cards is then ____________.

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374

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous Series

Let a,b ,c be positive integers such that is an integer. If a,b,c are in GP and the arithmetic mean of a,b,c, is b+2 then the value of is

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375

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Progression

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the firsteleven terms is 6: 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

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376

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If (a and b are positive real numbers) has 3 real roots, then prove that

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377

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

In internal angle bisector AI,BI and CI are produced to meet opposite sides in respectively. Prove that the maximum value of

is

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a1, a2, a3, , a100 a1 = 3andsp =p

∑i=1

ai, 1 ≤ p ≤ 100. n 1 ≤ n ≤ 20, m = 5n.

Sm

Sn

n, a2

n nk, k − 20 =

b

a

a2 + a − 14

a + 1

2x3 + ax2 + bx + 4 = 0 a + b ≥ 6(2 + 4 )13

13

ABC A' ,B' , C'

AI × BI × CI

AA' × BB' × C'

8

27

378

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

In how many parts an integer should be dissected so that the product of the parts is maximized.

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379

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

The minimum value of for positive real numbers is (a) (b) (c) (d)

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380

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

A rod of fixed length slides along the coordinates axes, If it meets the axes at , then the minimum value of

(a) (b) (c) (d)

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381

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

The least value of is 54 when . Is applicable for 54 when . Isapplicable for and18 is added further 78 when none

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382

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are in A.P. then the minimum value of is (a) (b) (c) (d)

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383

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If , then the least value of is (a) (b) (c) (d)

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N ≥ 5

x4 + y4 + z2

xyzx, y, z √2 2√2 4√2 8√2

k A(a, 0)andB(0, b)

(a + )2

+ (b + )2

1

a

1

b0 8 k2 + 4 +

4

k2k2 + 4 +

4

k2

6 tan2 φ + 54 cot2 φ + 18 (I) A.M. ≥ GM 6 tan2 φ, 54 cot2 φ, 18 (II) A.M. ≥ GM

6 tan2 φ, 54 cot2 φ (III) tan2 φ = cot2 φ (IV )

ab2c3, a2b3c4, a3b4c5 (a, b, c > 0), a + b + c 1 3 5 9

y = 3x−1 + 3−x−1 y 2 6 2/3 3/2

384

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Minimum value of (for real positive numbers is (a) (b) (c) (d)

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385

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If the product of positive numbers is , then their sum is (a) positive integer (b). divisible by (c)equal to (d)never less than

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386

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

The minimum value of when , is

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387

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are the three positive roots of the equation then the minimum value of equals

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388

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If positive numbers are in H.P., then equation has (a)both roots positive (b)both roots negative(c)one positive and one negative root (d)both roots imaginary

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389

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

For to have real solutions, the range of is

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(b + c)/a + (c + a)/b + (a + b)/c a, b, c) 1 2 4 6

n nn a n n + 1/n n2

P = bcx + cay + abz, xyz = abc

l,m, n x3 − ax2 + bx − 48 = 0, (1/ l) + (2/m) + (3/n) 1 2 3/25/2

a, b, c x2 − kx + 2b101 − a101 − c101 = 0(k ∈ R)

x2 − (a + 3)|x| = 4 = 0 a ( − ∞, − 7] ∪ [1, ∞) ( − 3, ∞) ( − ∞, − 7] [1, ∞)

390

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are the sides of a triangle, then the minimum value of is equal to (a) (b) (c) (d)

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391

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If then the minimum value of is (a) (b) (c) (d)none of these

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392

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If is always (a) (b) (c) (d)

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393

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If is always (a) (b) (c) (d)none of these

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394

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If , then the minimum value of is equal to (a) (b) (c) (d)

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395

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If and are in H.P., then (a) (b) (c) (d)none of these

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a, b, c + +a

b + c − a

b

c + a − b

c

a + b − c3 6 9 12

a, b, c, d ∈ R± 1, (log)da + (log)bd + (log)ac + (log)cb 4 2 1

a, b, c ∈ R+ , then + +bc

b + c

ac

a + c

ab

a + b≤ (a + b + c)

1

2≥ √abc

1

3≤ (a + b + c)

1

3≥ √abc

1

2

a, b, c ∈ R+ then(a + b + c)( + + )1

a

1

b

1

c≥ 12 ≥ 9 ≤ 12

a, b, c ∈ R+ a(b2 + c2) + b(c2 + a2) + c(a2 + b2) abc 2abc 3abc 6abc

a, b, c, d ∈ R+ a, b, c, d a + d > b + c a + b > c + d a + c > b + d

396

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If such that , then the maximum value of is equal to

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397

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

The minimum value of is

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398

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If then least value of is (a) (b) (c) (d)d. none of these

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399

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

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400

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive numbers is (a) (b) (c) (d) none of these

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401

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

For positive real numbers such that which one holds? (a) (b)

(c) (d) none of these

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a, b, c, d ∈ R+ a + b + c = 18 a2b3c4 218 × 32 218 × 33 219 × 32 218 × 33

|2z − 1| + |3z − 2|

a > 0, (a3 + a2 + a + 1)2

64a2 16a4 16a3

∫1

(1 − x2)√1 + x2

x, y, z A.P ., then y2 ≥ xz xy + yz ≥ 2xz + ≥ 4

x + y

2y − x

y + z

2y − z

a, bc a + b + c = p, (p − a)(p − b)(p − c) ≤ p38

27(p − a)(p − b)(p − c) ≥ 8abc

+ + ≤ pbc

a

ca

b

ab

c

402

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If first and terms of an AP, GP. and HP. are equal and their nth terms are a, b, c respectively, then (a) a=b=c (b)a+c=b (c) a>b>c and (d) none of these

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403

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

For , the smallest value of the function is

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404

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Let has two positive roots , then minimum value if is,

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405

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Miscellaneous

If are posirive real umbers and . The maximum value of equals.

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406

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive and then the maximum value of is (base of the logarithm is 10).

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407

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

Given that are positive real such that If the minimum value of is equal then the value of is.

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(2n − 1)thac − b2 = 0

x ≥ 0 f(x) = ,4x2 + 8x + 13

6(1 + x)

x2 − 3x + p = 0 aandb ( + )4

a

1

b

x, y, andz x =12 − yz

y + zxyz

a, b, andc 9a + 3b + c = 90, (log a + log b + log c)

x, y, z xyz = 32. x2 + 4xy + 4y2 + 2z2 m, m/16

408

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If satisfying then the maximum value of is.

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409

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

For any . Then the minimum value of is.

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410

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If are positive real numbers, then prove that (2004, 4M)

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411

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

The least value of the expression is a. 10 b. 2 c. ­0.01 d. none of these

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412

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

The product of positive numbers is unity. Then their sum is a. a positive integer b. divisible b equals to never less than

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413

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are different positive real numbers such that are positive, then is a. positive b. negative c. non­positive d. non­negative

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x, y ∈ R+ x + y = 3, x2y

x, y, ∈ R+ , xy > 0 + +2x

y3

x3y

3

4y2

9x4

a, b, c, (1 + a)(1 + b)(1 + c)7 > 77a4b4c4

2(log)10x − (log)x(0. 01), f or x > 1,

n n n + 1/n n

a, b, c b + c − a, c + a − b, anda + b − c(b + c − a)(c + a − b)(a + b − c) − abc

414

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

If are positive real umbers such that ,then satisfies the relation (a) (b) (c) (d)

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415

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Arithmetic Mean Of Mth Power

If are positive real numbers whose product is a fixed number then the minimum value of is is b.

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416

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Weighted Means

If is always greater than or equal to (a) (b) (c) (d)

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417

CENGAGE_MATHS_ALGEBRA_SEQUENCES AND SERIES_Inequalities Involving Simple A.M. G.M. H.M.

The minimum value of the sum of real number is

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a, b, c, d a + b + c + d = 2 M = (a + b)(c + d) 0 ≤ M ≤ 1 1 ≤ M ≤ 22 ≤ M ≤ 3 3 ≤ M ≤ 4

a1, a2, , an c, a1 + a2 + + an−1.......... . + 2anan−1 + 2an (n + 1)c1 /n 2nc1 /n (n + 1)(2c)1 /n

α ∈ (0, ), then√x2 + x +π

2

tan2 α

√x2 + x2 tanα 1 2 sec2 α

a−5, a−4, 3a−3, 1, a8, anda10witha > 0

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