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 :
LetS
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
,
.
.
.
.
.
.
∞