Q.Given the sum of an infinite geometric progression of real numbers is `S` ,which is a finite number.The sum of the squares of the terms is `S_1` .If the first term of the G.P. is 1,then :-

A) `S>S_1` ,if all the terms of G.P. is positive.

B) `S<S_1` ,if all the terms of G.P. is positive.

C) `S>S_1` ,if some of the terms of G.P. is negative.

D) `S<S_1` ,if some of the terms of G.P. is negative.

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Ans. A `S>S_1` if al terms of G.P. is positve

Because

Let

`1,r,r^2,r^3,........oo and r<1` then

`S=1/(1-r)`

another series is

`1,r^2,r^4,r^6,..........oo and r<1` which is alos in G.P. with common ratio `r^2` ,so sum is

`S_1=1/(1-r^2)`

Now

`r<1`

`r^2<r`

`Thus`

`r^2<r<1`

`Also`

`-r^2> -r`

`1-r^2>1-r`

`1/(1-r)>1/(1-r^2)`

`So`

`S>S_1`

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