Question

Let a 1, a 2, a 3,…. be an A.P. If
$\begin{array}{l} \displaystyle\sum\limits_{r=1}^\infty\frac{a_r}{2^r}=4,\end{array}$
then 4 a 2 is equal to ________.

Answer 1

Given,
$\begin{array}{l} S=\frac{a_1}{2}+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\frac{a_4}{2^4}+\cdots\infty\end{array}$
$\begin{array}{l} \frac{\frac{1}{2}S=~~\frac{a_1}{2^2}+\frac{a_2}{2^3}+\cdots\infty}{\frac{S}{2}~~=\frac{a_1}{2}+\frac{\left(a_2+a_1\right)}{2^2}+\frac{\left(a_3+a_2\right)}{2^3}}+\cdots\infty\end{array}$
$\begin{array}{l} \Rightarrow\ \frac{S}{2}=\frac{a_1}{2}+\frac{d}{2}\end{array}$
⇒ a 1 + d = a 2 = 4 ⇒ 4 a 2 = 16