Sequences and Series Session MPTCP05. 1.Revisit G.P. and sum of n terms of a G.P. 2.Sum of infinite terms of a G.P. 3.Geometric Mean (G.M.) and insertion

Sequences and Series

Session MPTCP05

1. Revisit G.P. and sum of n terms of a G.P.

2. Sum of infinite terms of a G.P.3. Geometric Mean (G.M.) and

insertion of n G.M.s between two given numbers

4. Arithmetico-Geometric Progression (A.G.P.) - definition, nth term

5. Sum of n terms of an A.G.P.6. Sum of infinite terms of an A.G.P.7. Harmonic Progression (H.P.) -

definition, nth term

Session Objectives

Geometric Progression

A sequence is called a geometric progression (G.P.) if the ratio between any term and the previous term is constant.

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The constant ratio, generally denoted by r is called the common ratio.

a1 = a

a2 = ar

a3 = ar2

a4 = ar3

an = ar(n-1)

First term

General Term

Problem Solving Tip

Choose Well!!!!

# Terms Common ratio

3 a/r, a, ar r

4 a/r3, a/r, ar, ar3 r2

5 a/r2, a/r, a, ar, ar2 r

6 a/r5, a/r3, a/r, ar, ar3, ar5 r2

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Important Properties of G.P.s

a, b, c are in G.P. b2 = ac _I005

Sum of n Terms of a G.P.

Sn = a+ar+ar2+ar3+ . . .+ar(n-1) ………(i)

Multiplying by r, we get,

rSn = ar+ar2+ar3+ . . .+ar(n-1)+arn ……...(ii)

Subtracting (i) from (ii), (r-1)Sn = a(rn-1)

n

n

r 1S a

r 1

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Sum of Infinite Terms of a G.P.

Sum of n terms of a G.P.,

n n n

n

r 1 1 r a arS a a

r 1 1 r 1 r 1 r

Case(i) r 1nar

as n , 01 r

n

nn n

a ar aS lim S lim

1 r 1 r 1 rCase(ii) r 1

naras n ,

1 r

n

nn n

a arS lim S lim

1 r 1 r

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aS , r 1

1 r

S , r 1

Illustration

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Single Geometric Mean

G is the G.M. of a and b

G2 = ab

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Geometric Mean – a Definition

If n terms G1, G2, G3, . . . Gn are inserted between two numbers a and b such that a, G1, G2, G3, . . . , Gn, b form a G.P.,

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then G1, G2, G3, . . . , Gn are called geometric means (G.M.s) of a and b.

Geometric Mean – Common Ratio

Let n G.M.s be inserted between two numbers a and b

The G.P. thus formed will have (n+2) terms.

Let the common ratio be r

Now b = ar(n+2-1) = ar(n+1)

1 mn 1 n 1

mb b

r , G aa a

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Property of G.M.s

Let n G.M.s G1, G2, G3, . . ., Gn be inserted between a and b.

Then,

n

n21 2 3 nG G G ...G G ab

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Illustrative Problem

Q. Insert 3 G.M.s between 4 and 9

A. Let the required G.M.s be G1,

G2 and G3.

Common ratio r =

1

49 3

4 2

13

G 4 2 62

23

G 4 62

33 3

G 4 3 62 2

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Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to

(a) (b)

(c) (d)

2 3

2 3

7 4 3

7 4 3

2

7 4 3

2

3

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A. Given that

Squaring both sides, we get,

a b2 ab

2

a b 4 ab

2 2a 2ab b 16ab Dividing by b2 and putting =r, we get,

ab

2r 14r 1 0 14 196 4

r2

r 7 4 3

_I008Q. If the A.M. between a and b is twice as

great as the G.M., a:b is equal to


Ans : (a)

A.22

22

2 3 2.2. 3r

2 3

22 3

r2 3 2 3

2 3 ar

b2 3

_I008Q. If the A.M. between a and b is twice as

great as the G.M., a:b is equal to

r 7 4 3

Arithmetico-Geometric Progression

A sequence is called an arithmetico-geometric progression (A.G.P.) if the nth term is a product of the nth term of an A.P. and the nth term of a G.P.

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a1 = a

a2 = (a+d)r

a3 = (a+2d)r2

a4 = (a+3d)ar3

an = {a+(n-1)d}r(n-1)

First term

General Term

Sum of n Terms of an A.G.P.

Consider an A.G.P. with general

term {a+(n-1)d}r(n-1).

Let the sum of first n terms be Sn

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

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

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

nn 1

n 2

a n 1 d ra 1 rS dr

1 r 1 r1 r

n 1n

n1 r

1 r S a dr a n 1 d r1 r

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Q. Find the sum of the first 10 terms of the given sequence :

1, 3x, 5x2, 7x3, . . .

A. Let

S = 1+3x+5x2+7x3+ . . . +{1+(10-1)2}x(10-1)

S = 1+3x+5x2+7x3+ . . . +19x9

xS = x+3x2+5x3+ . . . +17x9 +19x10

S-xS = 1+(2x+2x2+2x3+ . . . 2x9)-19x10

9

101 x

1 x S 1 2x 19x1 x

910

2

1 x1 19xS 2x

1 x 1 x

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Sum of Infinite Terms of an A.G.P.

Sum of n terms of an A.G.P.,

Case(i) r 1

nn 1

2 2

a n 1 d r1 r dras n , dr , 0

1 r1 r 1 r

nn 1

n 2n n

a n 1 d ra 1 rS lim S lim dr

1 r 1 r1 r

n 1 n

n 2

a 1 r {a (n 1)d}rS dr

1 r 1 r1 r

2a d r

S , r 11 r 1 r

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Sum of Infinite Terms of an A.G.P.

Case(ii) r 1

nn 1

2

a n 1 d r1 ras n , dr ,

1 r1 r

nn 1

n 2n n

a n 1 d ra 1 rS lim S lim dr

1 r 1 r1 r

S , r 1

Sum of n terms of an A.G.P.,

n 1 n

n 2

a 1 r {a (n 1)d}rS dr

1 r 1 r1 r

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Q. The sum to infinity of the series

is2 3

4 7 101 ...

5 5 5

(a) 16/35 (b) 11/8

(c) 35/16 (d) 8/6

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A. Let the required sum be S

Ans : (c)

2 3

4 7 10S 1 ...

5 5 5

2 3 4

S 1 4 7 10...

5 5 5 5 5

2 3

S 3 3 3S 1 ...

5 5 5 5

2

4S 3 1 1 3 5 71 1 ... 1 .

5 5 5 5 4 4535

S16

_I011Q. The sum to infinity of the series

is

2 3

4 7 101 ...

5 5 5


Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 + (1+b+b2+b3)r3 . . ., r and b being proper fractions is :

1

(a)1 r 1 br

1

(b)1 r 1 br

1

(c)1 r 1 br

1

(d)1 r 1 br

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A. Let the required sum be S

Subtracting, we have,

Ans : (a)

2 2 2 3 3S 1 1 b r 1 b b r 1 b b b r ...

2 2 3rS r 1 b r 1 b b r ...

2 2 3 3 4 41 r S 1 br b r b r b r ...

1

S1 r 1 br

_I011Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 + (1+b+b2+b3)r3 . . . (r and b being proper fractions ) is :

Harmonic Progression

A sequence is called a harmonic progression (H.P.) if the reciprocals of its terms form an A.P.

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11

aa

21

aa d

31

aa 2d

41

aa 3d

n1

aa n 1 d

First term

General Term

Class Exercise Q1.

Q. The first two terms of an infinite G.P. are together equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is :

1 1(a) 4, 1, , ....

4 16

4 8(b) 3, 2, , , ....

3 9

(c) Either (a) or (b)

(d) None of these

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Class Exercise Q1.

A. Let the first term of the G.P. be a and the common ratio be r.

Given that a+ar = 5 and

ar 1a 3 r

1 r 4

_I007Q. The first two terms of an infinite G.P. are

together equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is :

Now,

aa 5 a 4

4

Ans : (a)

Class Exercise Q2.

_I007Q. Find the value of p, if S for the G.P.

2

1 1 25p,1, , , . . . is

p 4p

Class Exercise Q2.

_I007Q. Find the value of p, if S for the G.P.

21 1 25

p,1, , , . . . isp 4p

A. S for the given G.P.

p 1S common ratio

1 p1p

2p 25

p 1 4

24p 25p 25 0

p 5 4p 5 0

5

Ans. p 5, or p4

Class Exercise Q3.

_I008Q. If one G.M. G and two A.M.s p and q are inserted between two quantities, show that G2 = (2p-q)(2q-p).

Class Exercise Q3.

_I008Q. If one G.M. G and two A.M.s p and q are

inserted between two quantities, show that G2 = (2p-q)(2q-p).

A. Let the two quantities be a and b.

a, p, q, b are in A.P.b a

3

Common difference =

b a 2a bp a

3 3

b a a 2bq a 2

3 3

2

R.H.S 2p q 2q p

4a 2b a 2b 2a 4b 2a b

3 3

ab

G

L.H.S.

Q.E.D.

Class Exercise Q4.

_I008Q. n G.M.s are inserted between 16/27 and 243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n.

Class Exercise Q4.

_I008Q. n G.M.s are inserted between 16/27 and

243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n.

A. Common ratio1 1 8

n 1 n 1 n 1b 243 27 3

a 16 16 2

Given that 8 n 1 32

2n 1 n 13 2 9 3

.2 3 4 2

8 n 5

2n 13 3

2 2

n 5 1n 7

n 1 4

Class Exercise Q5.

_I010Q. Find the sum of the series :

3 5 71 . . . n terms

2 4 8

Class Exercise Q5.


3 5 71 . . . n terms

2 4 8

A. We see that

n n 1

2n 1a

2

n n 1

3 5 7 2n 1S 1 . . .

2 4 8 2

nn 1 n

2n 1 2n 1S 1 3 5. . .

2 2 4 8 2 2

n n 1 n

3 2 2 2 2 2n 1S 1 . . .

2 2 4 8 2 2

Class Exercise Q5.


3 5 71 . . . n terms

2 4 8

n n 1 n

3 2 2 2 2 2n 1S 1 . . .

2 2 4 8 2 2

n 1

n n

11

3 1 2n 12S 1 2.

12 2 212

n 1

n n3 2 1 2n 1S 1 1

2 3 2 2

n n 1

3 1 1 2 2n 1S

2 3 3 22

n n 12 1 1 6n

S9 9 2

Class Exercise Q6.

_I010Q. Find sum to n terms of the series :

1+2x+3x2+4x3+ . . . (x 1)

Class Exercise Q6.

_I010Q. Find sum to n terms of the series :

1+2x+3x2+4x3+ . . . (x 1)

A. We see that an = nxn-1

Sn = 1+2x+3x2+4x3+ . . . +nxn-1

xSn = x+2x2+3x3+. . . +(n-1)xn-1+nxn

(1-x)Sn = 1+(x+x2+x3+ . . . xn-1)-nxn

n 1n

n

1 x1 x S 1 x nx

1 x

n n 1

n 2

1 nx 1 xS x

1 x 1 x

Class Exercise Q7.

_I011Q. Find the sum of infinite terms of the series :

2 3

4 6 82 . . .

11 11 11

Class Exercise Q7.

_I011A.

2 3

4 6 8S 2 . . .

11 11 11

2 3

S 2 4 6. . .

11 11 11 11

Q. Find the sum of infinite terms of the series :

2 34 6 8

2 . . .11 11 11

2 3

10S 1 1 12 2 . . .

11 11 11 11

10S 2 112

11 11 10 121

S50

Class Exercise Q8.


2 3

3 5 71 . . .

2 2 2

Class Exercise Q8.


2 3

3 5 71 . . .

2 2 2

A. 2 3

3 5 7S 1 . . .

2 2 2

2 3

S 1 3 5. . .

2 2 2 2

2 3

3 2 2 2S 1 . . .

2 2 2 2

3 1 2 1S 1 1

12 3 31

2

2

S9

Class Exercise Q9.

_I012Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then

(q-r)bc+(r-p)ca+(p-q)ab is equal to

(a) 1 (b) -1

(c) 0 (d) None of these

Class Exercise Q9.

_I012Q. If the pth, qth and rth terms of an H.P. be a,

b, c respectively, then


(a) 1 (b) -1

(c) 0 (d) None of theseA. The reciprocals the terms of the H.P. will be in A.P. Let this

A.P. have first term and common difference .

Given that

1

a ...(i)p 1

1

b ...(ii)q 1

1

c ...(iii)r 1

Class Exercise Q9.




(a) 1 (b) -1

(c) 0 (d) None of theseA. Taking reciprocal of (i), (ii) and (iii), we have

1p 1 ...(iv)

a

1q 1 ...(v)

b

1r 1 ...(vi)

c

Class Exercise Q9




(a) 1 (b) -1

(c) 0 (d) None of theseA. (iv)-(v), (v)-(vi), (vi)-(iv) gives,

b ap q ...(vii)

ab

c bq r ...(viii)

bc

a cr p ...(ix)

ac

Class Exercise Q9




(a) 1 (b) -1

(c) 0 (d) None of theseA. (vii)c, (viii)a and (ix)b gives,

bc cap q ab

ca abq r bc

ab bcr p ac

Adding,

ca ab ab bc bc caq r bc r p ca p q ab

Class Exercise Q9




(a) 1 (b) -1

(c) 0 (d) None of these ca ab ab bc bc caq r bc r p ca p q ab

A. (q-r)bc+(r-p)ca+(p-q)ab = 0

Ans : (c)

Class Exercise Q10.

Q. If ax = by = cz and x, y, z are in H.P. then a, b, c are in

(a) A.P. (b) H.P.

(c) G.P. (d) None of these

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Class Exercise Q10.

A. ax = by = cz = (say)

Ans : (c)

11 1yx za , b , c

x, y, z are in H.P.1 1 1y 2x 2z

11 1 1 1 2

2x 2z x zb .

b ac

_I012Q. If ax = by = cz and x, y, z are in H.P. then

a, b, c are in

(a) A.P. (b) H.P.

(c) G.P. (d) None of these

Documents

Sequences and Series Session MPTCP05. 1.Revisit G.P. and sum of n terms of a G.P. 2.Sum of infinite terms of a G.P. 3.Geometric Mean (G.M.) and insertion