#### 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 = aand common ratio = r

S = 4 ..... (given)

S = a / ( 1 − r) = 4

a=4(1-r)

a

^{3}=(1-r)

^{3}........(1)

Cubing each term of G.P.,

first term = a

^{3}

and common ratio = r

^{3}

Now the sum is

S ′ = a

^{3}/ ( 1 − r

^{3}) = 192

a

^{3}=192 ( 1-r)(1 + r+ r

^{2}).........(2)

Divide (2)by (1)

1=3(1 + r + r

^{2})/ (1 +r )

^{2}

( r

^{2}− 2 r + 1 ) = 3 ( r

^{2}+ r + 1 )

⇒ 2 r

^{2}+ 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 , . . . . . . ∞