Let a_1,a_2,a_3, be an A.P.

Question

Let $a_1,a_2,a_3,\ldots$ be an A.P. If $\displaystyle\sum_{r=1}^{\infty}\frac{a_r}{2^r}=4$, then $4a_2$ is equal to \_\_\_\_\_\_\_\_.

Answer: 16

Accepted Answer

Let $S=\sum\dfrac{a_r}{2^r}=4$. Then $\dfrac{S}{2}=\sum\dfrac{a_r}{2^{r+1}}$, and subtracting:
$$\frac{S}{2} = \frac{a_1}{2}+\frac{d}{2}\left(\frac{1}{2}+\frac{1}{4}+\cdots\right) = \frac{a_1}{2}+\frac{d}{2}.$$

So $a_1+d=4 \Rightarrow a_2=4 \Rightarrow 4a_2=16.$

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