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QUADRATIC EQUATIONS

Quadratic Expression

A polynomial of the form 2ax bx c , where , ,a b c are real or complex numbers and 0a , is

called a quadratic expression in variable ' 'x .

A complex number ' 'k is said to be a zero of quadratic expression 2ax bx c if 2 0ak bk c

Quadratic equations in one variable

Any equation of the form 2 0ax bx c where , ,a b c are real or complex numbers and 0a is

called a quadratic equation in the variable ' 'x . The numbers , ,a b c are called coefficients of the quadratic

equation.

C1 Solving Q.E and Relation between the Roots

Formula Method

To find roots of quadratic equation

Let 2 0ax bx c is given quadratic equation. Then

2 24 4 4 0a x abx ac (by multiplying with 4a both sides)

2 22 4ax b b ac 22 4ax b b ac

2 4

2

b b acx

a

The roots of the quadratic equation

2 0ax bx c are 2 4

2

b b ac

a

Let they are , and assume 2 24 4

,2 2

b b ac b b ac

a a

Then b

a ;

c

a

2 2 b cax bx c a x x

a a

2a x x = a x x

The quadratic equation having roots , is 0x x

or 2 0x x or 2 0x Sx P

Where S sum of roots, P product of roots.

Note1 An equation of the form 2 0ax bx c has more than two roots then it is an identity i.e.,

0, 0, 0a b c and it is true for all values of x

Note2 If i is a root of real quadratic equation then i is also a root.

Note3 If is a root of rational quadratic equation then is also a root where 2k .

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Practice Problems

1. Find the roots of the equation

(a) 2 2 5 0x x (b) 2 2 24 0x x (c) 2 4 13 0x x

2. Find the quadratic equation with leading coefficient one whose roots are

(a) 2 3 and 2 3 (b) 1 2i and 1 2i

3. Find a quadratic equations with real coefficients for which one of the root is 2 3i

4. Find a quadratic equation whose roots are 2 3i and 2

5. Solving the following equations

i) 4 25 6 0x x ii) 2/3 1/3 2 0x x iii) 1 13 3 10x x

iv) 1 14 3.2 2 0x x v)

21 1

2 7 5 0x xx x

, 0x

vi)

22

1 15 6 0x x

xx, 0x vii)

21 3 1

42

x xx x

viii) 1 3 5 7 9x x x x (Hint: 21 : 8M x x t ; 2 : 4M x t )

ix) 4 3 22 11 2 0x x x x x)

3 5

3 2

x x

x x, 0x

6. Solve the following

i) 3 1 1 2x x ii) 2 1 3 2 5 3x x x iii) 1 2 5 3x x

7. Solve the following

i) 3 8 2x ii) 2 6 5 8 2x x x iii) 22 8x x

iv) 2 26 6

2 5 4

x x x x

x x

vi) 2 3 10 8x x x

8. If , are the roots of 2 2 4 0x x find

(a) 1 1

(b) 4 4 (c)

1 1

(d) 3 3

1 1

(e) 5 5 (f) 6 6

9. Let and 2 3 5 and 2 5 3 then find the quadratic equation whose roots are

and

10. If one root of 2 0x x k is square of the other then find the value of ' 'k .

11. If and are the roots of 2 0ax bx c then find

(a) 1 1

a b a b

(b)

a b a b

(c) 2 2

a b a b

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12. If and are the roots of 2 0ax bx c find the equation whose roots are

(a) 1 1 1

,

(b) 1

, 1

(c) 2 22 2

1 1,

13. Solve the equation

1x b x c x c x a x a x b

a b a c b c b a c a c b

14. If , are the roots the equation 2 1 0x px and , are the roots of the equation 2 1 0x qx then find the value of

15. Given that , be the roots of 2 4 1 0Ax x and , are the roots of 2 6 1 0Bx x , find

the values of A and B such that , , , are in H.P

Answer & Key

1. (a) 1 6 (b) 4, 6 (c) 2 3 , 2 3i i

2. (a) 2 4 1 0x x (b) 2 2 5 0x x

3. 2 4 13 0x x 4. 2 4 3 4 6 0x i x i

5. i) 2 , 3 ii) 8,1 iii) 1, 1 iv) 1,2

v) 1 1 3

2, ,2 2

i vi)

1 32 3,

2

i vii)

11, ,1,2

2 viii) 4,4,4 10

ix) 1 3 5

,2,2 2

x) 1,4

6. i) 5,1x ii) 1

2x iii) 3x

7. i) x ii) 3,5x iii) 2,2 2x iv) 2, 1 3x

v) 74

,13

x

8. (a) 1

2 (b) 16 (c)

3

2 (d)

1

4 (e) 32 (f) 52

9. 19

3 10. 2 5 11. (a)

b

ac (b)

2

a (c)

2

2 2

2b ac

a c

12. (a) 2 0bcx a b x ab (b) 22 0acy a c by a c

(c) 22 2 2 2 2 22 2 0c x b ac c a x b ac

13. Every complex number 14. 2 2q p

15. 21, 16A B

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C2 Nature of roots

For the quadratic equation 2 0ax bx c , 2 4b ac is called the discriminant of the quadratic

equation denoted by D or .

2 4D b ac

If , ,a b c are real then the nature of roots of equation is as follows

1. If 2 4 0b ac , then the roots of the equation are real and equal

2. If 2 4 0b ac , then the roots of the equation are real and distinct

3. If 2 4 0b ac , then the roots of the equation are non real conjugate complex numbers

Practice Problems

1. Find the nature of the roots of the equation

(a) 22 3 7 0x x (b) 22 3 0x x

2. If 2 2 1 3 7 3 2 0x m x m has equal roots then find the value of ' 'm .

3. Prove that the biquadratic equation 2ax bx c 2 0ax bx c has at least two real roots.

4. If the roots of the equation 2 28 6 0x x a a are real and distinct then find the range of ' 'a .

5. If 0a b c , then find the nature of the roots of the equation 24 3 2 0, 0ax bx c a

6. Show that if , , ,p q r s are real numbers and 2pr q s then atleast one of the equations

2 0x px q , 2 0x rx s has real roots

7. Prove that the roots of the equation 0x a x b x b x c x c x a are always

real and they will be equal if and only if a b c

8. If the roots of the equation 2 0a b c x b c a x c a b are equal, prove that , ,a b c are in

H.P

9. If ,a b c d then prove that for every real , the quadratic equation

0x a x c x b x d has real roots.

10. If a and b are odd integers, then find the number of real roots of 2 0x a x b , where .

denotes ‘Greatest Integer Function’.

Answer & Key

1. (a) non real complex (b) real and distinct 2. 10

2,9

4. 2,8a 5. Real & distinct 10. Zero

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C3 Graph & Sign changes of a Quadratic Expression

The graph of 2y ax bx c is as follows

(i) 0, 0a D (ii) 0, 0a D

V

V

(iii) 0, 0a D (iv) 0, 0a D

V

V

(v) 0, 0a D (vi) 0, 0a D

V

V

The value of V in each figure is 24

,2 4

b ac b

a a

Practice Problems

1. If 2 0ax bx c has imaginary roots and a c b , then prove that 4 2a c b .

2. If the equation 2 2 0ax bx c has non real roots then prove that

2 2 22a c b .

3. If 2 6 0ax bx doesn’t have distinct real roots then find the least value of 3a b

4. Let 2f x ax bx c be a quadratic expression having its vertex at 3, 2 and the value of

0 7f then find f x .

5. Find the sum of the abscissa of the points where the curves 23 5 8 2 5y kx k x k ,

k touch the x-axis

6. If 2,2020 is the highest point on the graph of 22 4y x ax k then find ' 'k

7. Draw the graphs of the following

i) 2 5 6f x x x ii) 22 12 18f x x x iii) 23 3 1f x x x

iv) 2 3 4f x x x v) 2 10 25f x x x vi) 22 2 2f x x x

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8. Discuss the sign changes of the following Quadratic Expressions, where x is real

i) 2 5 4x x ii) 2 3x x iii) 23 4 4x x

iv) 25 4 2x x v) 2 7 10x x vi) 215 4 3x x

9. Solve the following inequalities

i) 215 4 4 0x x ii) 2 2 1 0x x iii) 2 4 21 0x x

10. If 2f x ax bx c , 0a , , ,a b c then write necessary and sufficient conditions for the

following statements

i) 0,f x x ii) 0,f x x

iii) 0,f x x iv) 0,f x n

Answer & Key

3. 2 4. 2 6 7f x x x 5. C 6. 2012

PROGRESSION

C1 Arithmetic Progression (A.P)

A sequence of terms 1 2,t t , 3...., ,....nt t are said to be in A.P if

1k kt t constant k . The first term 1' 't in an A.P is usually denoted by ' 'a and the

constant difference between two successive terms by ' 'd called common difference of the A.P. Therefore

the A.P is of the form , , 2 , 3 ,.....a a d a d a d

nth term of an A.P

The nth term of an A.P denoted by ' 'nt is given by

1nt a n d

Note: Since ' 'a and ' 'd are constants and 1nt a n d , the nth term of an A.P is linear expression in

' 'n

Practice Problems

1. In an A.P if 5 3nt n then find ' 'd .

2. In an A.P of m nmt nt then find m nt

3. If 1

mtn

and 1

ntm

then find mnt

4. In an A.P if 4

7

2

3

t

t then find 6

8

t

t

5. Find the number of identical terms in the two AP’s

2, 5, 8, 11, …….. 60 terms and 3, 5, 7, 9, ….. 50 terms

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6. The interior angles of a polygon are in arithmetic progression. The smallest angle is 120 and the

common difference is 5 . Find the number of sides of the polygon.

7. If 1 2 3, , ,.....,a a a na are non zero terms which are in A.P then prove

1 2 2 3 1 1

1 1 1 1......

n n n

n

a a a a a a a a

8. In an A.P if 5 717, 15t t then find 22t

C2. Sum of ‘n’ terms of an A.P

The sum of first ‘n’ terms of an A.P is denoted by nS and is given by

2 ....... 1nS a a d a d a n d

1 2 3 ..... 1na d n

1

2

n nna d

2 12

na n d

first term + Last term2

n

Note: nS is of the form 2an bn where ,a b are constants

Practice Problems

1. In an A.P if 23 4 ,nS n n find 10t

2. In an A.P if ,m nS n S m then find m nS

3. Find the sum of all three digit natural numbers which are divisible by 7

4. Find the sum of integers from 1 to 100 which are divisible by either 2 or 5

5. In an A.P if 1 20t , pt q , qt p find the value of ' 'm such that sum of the first ' 'm terms of the

A.P is zero

6. Find the number of terms of an A.P series 1 2

20, 19 ,18 ,3 3

….. of which the sum of terms is 300.

7. If the ratio of the sum of ' 'n terms of two AP’s is 7 1

4 27

n

n

, then find the ratio of their 12th terms.

8. If 1 2 3, ,S S S are the sum of first ,2 ,3n n n terms respectively of an A.P then show that

3 2 13S S S .

9. If the sum of first 8 and 19 terms of an A.P are 64 and 361 respectively, find the common

difference and nS .

10. Let nS denote the sum of first ' 'n terms of an A.P. If 2 3n nS S then the find the ratio 3n

n

S

S

11. Find the least value of ' 'n for which the sum 2 5 8 .... n terms exceeds 1000.

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12. Find the maximum value of the sum of the series 1 2

10 9 8 ......3 3

13. The number of terms of an A.P is even, the sum of odd terms is 24, of even terms is 30, and the

last term exceeds the first term by 21

2 then find the number of terms

14. If 1 2, ,.....SmS S are the sum of ' 'n terms of m AP’s whose first terms are 1,2,3,…..,m and the

common differences are 1,3,5,…… 2 1m . Then find 1 2 ....... mS S S

15. The set of natural numbers is divided into sets 1 1S , 2 2,3S , 3 4,5,6S ,

4 7,8,9,10S and soon. Then find the sum of elements in the set 50S .

C3. Properties of an A.P

1. If 1 2 3 2, , ,....., na a a a are in A.P then

(a) 1 2 2 2 1 1.....n n n na a a a a a

(b) , 0,1,2,....,2

n k n kn

a aa k n

2. If three terms , ,a b c are in A.P then 2b a c

3. (a) Three terms in A.P are taken as , ,a d a a d

(b) Four terms in A.P are taken as 3 , , , 3a d a d a d a d

4. If 1 2 3, , ,...., na a a a are in A.P then

(a) 1 2 3, , ,......, na k a k a k a k are in A.P k

(b) 1 2 3, , ,.....,ana k a k a k k are also in A.P k

5. If 1 2 3, , ,....., na a a a and 1 2 3, , ,....., nb b b b are two A.P’s then

1 1 2 2 3 3, , ,......, n npa qb pa qb pa qb pa qb are also in A.P ,p q

Practice Problems

1. If , ,a b c are in A.P, prove that the following are also in A.P

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

, ,bc ca ab

(c) 1 1 1

, ,a b cbc ca ab

(d) 1 1 1 1 1 1

, ,a b cb c c a a b

(e) 2 2 2, ,bc a ca b ab c (f) 2 2 2, ,a b c b c a c a b

(g) 1 1 1

, ,b c c a a b

(h) 2 2 22 2 2, ,b c a c a b a b c

2. If 2 2 2, ,b c c a a b are in A.P, then prove that

1 1 1, ,

b c c a a b are also in A.P

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C4. Arithmetic Means (A.M’s)

Let ' 'a and ' 'b are two quantities. If 1 2, ,...., nx x x are ' 'n quantities such that 1 2, , ,......, ,na x x x b are

in A.P then 1 2, ,...., nx x x are called ' 'n arithmetic means (A.M’s) between ' 'a and ' 'b

Here 2 1nb t a n d

common difference 1

b ad

n

Sum of ' 'n A.M’s

1 2 1......2

n n

nx x x x x

2

na b

Practice Problems

1. Between ‘1’ and ‘31’ are inserted ' 'm arithmetic means so that the ratio of the 7th and 1m th

means is 5 : 9. Find the value of m .

2. The sum of two numbers is 13

6. An even number of AM’s are being inserted between them and

their sum exceeds their number by 1. Find the number of means inserted.

3. There are ' 'n AM’s between 3 and 17. If the ratio of the first mean to last mean is 1 : 3, find the value of ' 'n .

4. ' 'n arithmetic means are inserted in between x and 2y and then between 2x and y . In case the

rth means in each case be equal then find the ratio x

y.

5. Between two numbers whose sum is 1

26

, an even number of A.M’s are inserted. If the sum of

these means exceeds their number by unity, then find the numbers means

6. If , ,a b c are in A.P and ' 'P is the A.M between a and b and ' 'q is the A.M between b and c ,

then find the A.M between p and q .

C5. Geometric Progression (G.P)

A sequence of terms 1 2 3, , ,......... ,......nt t t t are said to be in G.P if 1k

k

t

t constant k .

The first term 1' 't is denoted by ' 'a and the constant ratio between two successive terms by ' 'r

called common ratio.

Then a G.P is of the form 2, , ,......a ar ar

Note: All the terms in the G.P are non zero.

nth term of a G.P

The ' 'n th term of a G.P denoted by ' 'nt is given by 1. nnt a r

Sum of ‘n’ terms of a G.P

The sum of ' 'n terms of G.P denoted by nS is given by

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2 1........ nnS a ar ar ar

, 1

1, 1

1

n

na if r

ra if r

r

If 1 1r then nr tends to zero for infinite value of n

In this case the sum of infinite terms denoted by , 1 11

aS r

r

Practice Problems

1. In a G.P if 5 81t , 2 24t then find 8S

2. In a G.P if 3 6: 125 : 152S S then find ' 'r

3. In a G.P, 255nS , 128nt , 2r then find ' 'n

4. A G.P consists of an even number of terms. The sum of all the terms is three times that of odd terms. Determine the common ratio of the G.P.

5. In an increasing G.P, the sum of the first and last term is 66, the product of the second and last but one term is 128, and the sum of all the terms is 126. Find the number of terms

6. If 1 2 3, ,S S S be respectively the sums of ,2 ,3n n n terms of a G.P then prove that

2 21 2 1 2 3S S S S S

7. If 1 2, ,......, nS S S are the sums of infinite geometric series whose first terms are 1,2,3,….,n and

common ratio’s are 1 1 1

, ,2 3 4

,…..,1

1n respectively then find

1) 1 2 3 ...... nS S S S 2) 2 2 2 21 2 3 ........ nS S S S

8. Find 2 2 31 1 1 1 .......x x x x x x n terms

9. If 21 .......x a a

21 ......y b b

Where 1, 1a b , prove that 2 21 ......1

xyab a b

x y

10. Given 1 2 3, ,t t t are in G.P and 1 2 3, , 64t t t are in A.P and 1 2 3, 8, 64t t t are in G.P then find

1 2 3, ,t t t

C6. Properties of G.P

1. If 1 2 3 2, , ,......, na a a a are in G.P then

(a) 1 2 2 2 1 3 2 2 1.........n n n n na a a a a a a a

(b) 2 0n n k n ka a a k to 1n

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2. If three terms , ,a b c are in G.P then 2b ac

3. (a) Three terms in G.P are taken as , ,aa ar

r

(b) Four terms in G.P are taken as 33

, , ,a a

ar arrr

4. If 1 2 3, , ,....., na a a a are in G.P then

(a) 1 2 3, , ,....., na k a k a k a k are also in G.P 0k

(b) 1 2 3, , ,.......,k k k kna a a a are also in G.P k R

5. If 1 2 3, , ,......, na a a a and 1 2 3, , ,....., nb b b b are in G.P then

1 1 2 2 3 3, , ,.......,p q p q p p p qn na b a b a b a b are also in G.P ,p q

Practice Problems

1. If , , ,a b c d are in G.P, prove that 2 2 2 2 2 2, ,a b b c c d are in G.P

2. If three successive terms of a G.P with common ratio 1r r form the sides of a ABC then

find r r (where . denotes the G.I.F)

3. Find the number of increasing G.P(s) with first term unity, such that any three consecutive terms, on doubling the middle become an A.P

4. If , , ,a b c d are in G.P, then find

2 2 2 2 2 2

2

a b c b c d

ab bc cd

5. A G.P consists of even number of terms. If the sum of terms of occupying the odd places is 1S

and that of terms in the even places is 2S . Then find the common ratio of G.P

C7. Geometric Means (GM’s)

Let ' 'a and ' 'b are two quantities. If 1 2, ,...., nx x x are ' 'n quantities such that 1 2 3, , , ,...., ,na x x x x b

are in G.P then 1 2 3, , ,......, nx x x x are called ' 'n G.M’s between ' 'a and ' 'b

Now 1

2 . nnb t a r

Common ratio

1

1nbr

a

Product of ' 'n GM’s between a & b

21 2 3.....n

nx x x x ab

Practice Problems

1. If one GM ' 'G and two AM’s ' 'p and ' 'q are inserted between two given numbers, then show

that 2 2 2G p q q p

2. If two geometric means 1g and 2g and one arithmetic mean ' 'A be inserted between two

numbers, then find 2 21 2

2 1

g g

g g

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3. If p p

q q

a b

a b

is G.M between a and b and 1p q , then find the value of ' 'p .

4. If , ,a b c are three distinct real numbers in G.P such that 2a b c xb , then find range of ' 'x

5. Let ' 'S be the sum, ' 'P be the product and ' 'R be the sum of reciprocals of ' 'n terms of G.P the find ' 'S in terms R and ' 'P

C8. Harmonic Progression (H.P)

A sequence of terms 1 2 3, , ,....., nt t t t are said to be in H.P if 1 2 3

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

nt t t t are in A.P

Therefore an Harmonic progression is of the form 1 1 1

, , ,......2a a d a d

nth term of H.P

The nth term of an H.P denoted by nt is given by

1

1nt

a n d

Practice Problems

1. The 8th and 14th term of a H.P are 1

2 and

1

3 respectively. Find its 20th term

2. The m th term of a H.P is ' 'n and the n th term is ' 'm . Prove that r th term is mn

r

3. An A.P a G.P and a H.P have a and b for their first two terms. If their 2nd

n terms are in G.P,

then find

2 2 2 2

2 2

n n

n n

b a

ab b a

4. If 1 2 3, , ,....... na a a a are in H.P, then prove that 1

2 3 ..... n

a

a a a , 2

1 3 ..... n

a

a a a ,

…….1 2 1.....

n

n

a

a a a are in H.P.

5. If positive numbers , ,a b c are in H.P, then find the least value of 2 2

a b c b

a b c b

C9. Harmonic Mean (H.M)

If three quantities , ,a x b are in H.P then ' 'x is called H.M of ' 'a and ' 'b

1 1 1

, ,a x b

are in A.P

2 1 1

x a b

2ab

xa b

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If 1 2, , ,......, ,na x x x b are in H.P then 1 2, ,......, nx x x are called ' 'n harmonic means (H.M’s) between

' 'a and ' 'b . In this case

1 11

1

a bn d d

b a n ab

Practice Problems

1. If ' 'H be the H.M between ' 'a and ' 'b then prove that 1 1 1 1

H a H b a b

2. The sum of three numbers in H.P is 26 and sum of their reciprocal is 3

8. Find the numbers

3. If , ,a b c are in H.P, prove that 2b a b c

b a b c

4. The value of x y z is 15 if , , , ,a x y z b are in A.P, while the value of 1 1 1

x y z is

5

3 if

, , , ,a x y z b are in H.P. Find a & b

5. If 1 2 1 2 1 2, ; , ; ,A A G G H H be two AM’s ; GM’s ; HM’s respectively of ' 'a and ' 'b then prove that

1 2 1 2

1 2 1 2

G G A A

H H H H

6. If 1 2 3, , ,....., na a a a are in H.P prove that 1 2 2 3 1 1....... 1n n na a a a a a n a a

7. If 1 2 3, ,a a a are in A.P; 2 3 4, ,a a a are in G.P and 3 4 5, ,a a a are in H.P prove that 1 3 5, ,a a a are in G.P.

8. If , ,a b c are in H.P then prove that

(a) , ,a b c

b c c a a b are in H.P (b) , ,

a b c

b c a c a b a b c are in H.P

(c) , ,bc ca ab

b c c a a b are in A.P

9. Consider two positive numbers a and b . If A.M of a and b exceeds their G.M by 3

2 and G.M of

a and b exceeds their H.M by 6

5, then find 2 2a b

10. Let ' 'P be the first of ' 'n A.M’s between two numbers a and b and ' 'q is the 1st of ' 'n H.M’s

between the same numbers. Find the internal in q

p lies

C10. Arithmetico-Geometric sequence

A sequence of the form 2 3, , 2 , 3 ,......a a d r a d r a d r is called Arthmetico-Geometric

sequence.

The n th term of the above sequence 11 nnt a n d r

and the sum of ' 'n terms nS is given by

2 12 ....... 1 nnS a a d r a d r a n d r

2 32 ...... 1 nnrS ar a d r a d r a n d r

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Subtracting

2 1....... 1n nn n rS rS a dr dr d a n d r

11

1 11

nn

n

rr S a dr a n d r

r

1

2

11

1 11

nn

n

a n d ra rS dr

r rr

If 1 1r then the sum of infinite Arithmetico-Geometric series is 21 1

a drS

r r

Practice Problems

1. Find the sum of infinity of the series 2 31 3 5 7 ......., 1x x x x

2. Find the sum to infinity of the series 2 2 2 2 2 31 2 3 4 .......x x x

3. Find the sum to n terms of the seirs 3 5 9 17 ......

4. Find sum to infinity of the series 2 5 2 11

........3 6 3 24

5. Find the sum of first ‘n’ terms of the series 2 2 2 2 2 21 2 2 3 2 4 5 2 6 ......

C11. ∑ Notation

1 2

1

.....n

r n

r

T T T T

where nT is the general term

Eg1:- 4

2 2 2 2 2

1

1 2 3 4r

r

Eg2:- , ,a b c

a b c a b c b c a c a b

Practice Problems

1. Find 1

n

r

r or n 2. Find 2

1

n

r

r or

2n

3. Find 1

3n

r

r or

3n 4. Find

1

1n

r

r r

5. Find 1

1

1 2

n

rr r r

HYDERABAD Centres: Saifabad, Kukatapally, Dilsukhnagar, Narayanaguda, Madhapur, Miyapur - 15 - Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942

ANSWER & KEY

Concept-1

1. 5 2. 0 3. 1 4. 4

5

5. 17 6. 9 8. 0

Concept-2

1. 61 2. m n 3. 70336 4. 3050

5. 41 6. 25 or 36 7. 162

119 9. 22, nd S n

10. 6 11. 26 12. 80 13. 8

14. 1

2

mn mn 15. 62525

Concept-4

1. 14 2. 12 3. 6 4. 1

r

n r

Concept-5

1. 8 83 2

8

2.

3

5 3. 8 4. 2

5. 6 7. (1) n (2) 2 1n

8.

2

1 1

1

nx x n x

x

10. 4, 20, 100

Concept-6

2. –1 3. 1 4. 1 5. 2

1

S

S

Concept-7

2. 2A 3. 1

2 4.

1 2, ,

2 3

5. 2/nR P

Concept-8

1. 1

4 3.

1n

n

5. 4

Concept-9

2. 6, 8, 12 4. 1, 9a b 9. 159

10. q

p does not lie between 1 and

21

1

n

n

HYDERABAD Centres: Saifabad, Kukatapally, Dilsukhnagar, Narayanaguda, Madhapur, Miyapur - 16 - Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942

Concept-10

1. 2

1

1

x

x

2.

3

1

1

x

x

3. 12 2n n 4.

2

9

5. 2

1

2

n n if n is even

2 1

2

n n if n is odd

Concept-11

1. 1

2

n n 2.

1 2 1

6

n n n 3.

2

1

2

n n or

22 1

4

n n

4. 1 2

3

n n n 5.

3

4 1 2

n n

n n