If the sum of an infinite GP a, ar, ar2 , ar3 , ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2 , ar4 , ar6 , ....... is :
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Explanation Sum of infinite terms : $${a \over {1 - r}} = 15$$ ..... (i) Series formed by square of terms : a2 , a2 r2 , a2 r4 , a2 r6 ....... 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 = $${{a{r^2}} \over {1 - {r^2}}} = {{12.\left( {{1 \over {25}}} \right)} \over {1 - {1 \over {25}}}} = {1 \over 2}$$