Question

The sum of infinite terms of a geomteric series is 3/2. The sum of the 1st, 3rd, 5th, 7th...terms of the same series is 9/8
Find the common ratio of the GP
(1) 1. (2) 1/3. (3) 1 or 1/3. (4) cannot be determined. (Ans is 1/3)

Answer 1

Answer:

Let the first term be a and the common ratio be r.

The first sum is

1−r

a

=2

Now, the terms have been cubed, so the sum becomes

1−r

3

a

3

=24

Substituting a to be 2(1−r) in the second equation, we get 8(1−r)

3

=24(1−r

3

)

⇒1−3r+3r

2

−r

3

=3−3r

3

⇒2r

3

+3r

2

−3r−2=0

⇒(r−1)(2r

2

+5r+2)=0

So, r=1,−0.5,−2

r cannot be 1,and also −2 is not allowed since the terms have to reduce, so r is −0.5, we get a=2(1−r)=3