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 , . . . . . . ∞