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Q.
If the sum of an infinite GP $a, ar, ar^2, ar^3 ,...$ is $15$
and the sum of the squares of its each term is $150$,
then the sum of $ar^2, ar^4, ar^6, ...$ is :
Sum of infinite terms:
$\frac{ a }{1- r }=15 \,\,\, ....... (i)$
Series formed by square of terms:
$a ^{2}, a ^{2} r ^{2}, a ^{2} r ^{4}, a ^{2} r ^{6} \ldots$
Sum $=\frac{ a ^{2}}{1- r ^{2}}=150$
$\Rightarrow \frac{ a }{1- r } \cdot \frac{ a }{1+ r }=150 $
$\Rightarrow 15 \cdot \frac{ a }{1+ r }=150$
$\Rightarrow \frac{ a }{1+ r }=10 \ldots \ldots \ldots $ (ii)
by (i) and (ii) $a =12 ; r =\frac{1}{5}$
Now series : $a r^{2}, a r^{4}, a r^{6}$
Sum $=\frac{ ar ^{2}}{1- r ^{2}}=\frac{12 \cdot(1 / 25)}{1-1 / 25}=\frac{1}{2}$