JEE MAIN - Mathematics (2022 - 29th July Morning Shift - No. 18)
Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an A.P. If $$\sum\limits_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4$$, then $$4 a_{2}$$ is equal to _________.
Answer
16
Explanation
Given
$$S = {{{a_1}} \over 2} + {{{a_2}} \over {{2^2}}} + {{{a_3}} \over {{2^3}}} + {{{a_4}} \over {{2^4}}}\, + \,.....\,\infty $$
$${{{1 \over 2}S = {{{a_1}} \over {{2^2}}} + {{{a_2}} \over {{2^3}}}\, + \,.........\,\infty } \over {{S \over 2} = {{{a_1}} \over 2} + {{({a_2} + {a_1})} \over {{2^2}}} + {{({a_3} + {a_2})} \over {{2^3}}}\, + \,......\,\infty }}$$
$$ \Rightarrow {S \over 2} = {{{a_1}} \over 2} + {d \over 2}$$
$$ \Rightarrow {a_1} + d = {a_2} = 4 \Rightarrow 4{a_2} = 16$$
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