Question

Find the sum: $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}.....$ up to $n$ terms

Answer 1

$S_n=\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+...$ up to $n$ terms

$S_n=\sqrt{2}+\sqrt{4\times 2}+\sqrt{9\times 2}+\sqrt{16\times 2}+...$up to $n$ terms

$=\sqrt{2}+2\sqrt{2}+3\sqrt{2}+4\sqrt{2}+...$up to $n$ terms.

$=\sqrt{2}(1+2+3+4+...)$ up to $n$ terms.

$=\sqrt{2}\frac{n(n+1)}{2}$ where sum of all natural numbers$=\frac{n(n+1)}{2}$

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