Question :

The sum of an infinite number of terms of a G.P. is 4 , and the sum of their cubes is 192 ; find the series.
Submitted on 8/7/2024 | Answered by Vandana Rana

Answer :

Let
S
be the sum of infinite terms of G.P. having first term
=
a

and common ratio
=
r


S
=
4
..... (given)

S
=
a / (
1

r)
=
4
a=4(1-r)

a3=(1-r)3........(1)

Cubing each term of G.P.,
first term
=
a3


and common ratio
=
r3



Now the sum is


S

=
a3/
(
1

r 3)

=
192


a3 =192 ( 1-r)(1 + r+ r2).........(2)

Divide (2)by (1)

1=3(1 + r + r2)/
(1 +r )2







(
r2




2
r
+
1
)
=
3
(
r2

+
r
+
1
)



2
r2

+
5
r
+
2
=
0



r
=

2
or

1/
2


r
=

2
will be rejected as above formulae is valid only for
|
r
|
<
1



r
=

1/
2



a
=
6


Thus, the series is

6
,

3
,
3
2
,
.
.
.
.
.
.