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