Question

Sum to n terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+...$ is

• A. $\frac{n(n+1)}{2}$
• B. $2n(n+1)$
• C. $\frac{n(n+1)}{\sqrt{2}}$
• D. $1$

Answer 1

Answer: (C)

$\frac{n(n+1)}{\sqrt{2}}$

Given, $S_n$ $=$ $\sqrt{2}$ $+$ $\sqrt{8}$ $+$ $\sqrt{18}$ $+$ $\sqrt{32}$ $+.......$
$=$ $\sqrt{2}$ $+$ $\sqrt{2.4}$ $+$ $\sqrt{2.9}$ $+$ $\sqrt{2.16}$ $+.......$
$=$ $\sqrt{2}$ $(1+$ $\sqrt{4}$ $+$ $\sqrt{9}$ $+$ $\sqrt{16}$ $+.......)$
$=$ $\sqrt{2}$ $(1+2+3+4+.......)$
$=$ $\sqrt{2}\times$ $\frac{n}{2}(n+1)$
$=$ $\frac{n}{\sqrt{2}}$ $(n+1)$