Official solution
Sum of infinite terms :
\[{a \over {1 - r}} = 15\] ..... (i)
Series formed by square of terms :
a2, a2r2, a2r4, a2r6 .......
Sum = \[{{{a^2}} \over {1 - {r^2}}} = 150\]
\[ \Rightarrow {a \over {1 - r}}.{a \over {1 + r}} = 150 \Rightarrow 15.{a \over {1 + r}} = 150\]
\[ \Rightarrow {a \over {1 + r}} = 10\] ...... (ii)
by (i) and (ii), a = 12; r = \[{1 \over 5}\]
Now, series : ar2, ar4, ar6
Sum = \[{{a{r^2}} \over {1 - {r^2}}} = {{12.\left( {{1 \over {25}}} \right)} \over {1 - {1 \over {25}}}} = {1 \over 2}\]