Sequences and Series Part 2

We have, sum of infinite GP a,ar,ar2,....... is
S∞=

a

1−r

=15 ...(i)
and sum of infinite GP a2,a2r2,a2r4,..... is
S′∞=

a2

1−r2

=150
⇒(

a

1−r

)(

a

1+r

)=150 ...(ii)
Divide Eq. (ii) by Eq. (i)

a

1+r

=10 ...(iii)
Divide Eq. (iii) by Eq. (i)

1−r

1+r

=

10

15

=

2

3

⇒3−3r=2+2r
⇒1=5r
⇒r=

1

5

Now, putting r=

1

5

in Eq. (iii), we get

a

1+ 1 5

=10
⇒

5a

6

=10
⇒ a = 12
Now, sum of ar2,ar4,ar6,......,∞
S′′∞=

ar2

1−r2

=

12.( 1 25 )

24 25

=

1

2