Question

Find the value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+...........+\frac{1}{1+2+3+\mathrm{4...}n}$

Answer 1

$1+\frac{1}{1+2}+\frac{1}{1+2+3}+...........+\frac{1}{1+2+3+\mathrm{4...}n}$

$1+\frac{2}{2\times 3}+\frac{2}{3\times 4}+.......+\frac{2}{n\times (n+1)}$

$1+2(\frac{3-2}{2\times 3}+\frac{4-3}{3\times 4}+..............+\frac{n+1-n}{n\times (n+1)})$

$1+2(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}......................\frac{1}{n}-\frac{1}{n+1})$

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

$2-\frac{2}{n+1}$

$\frac{2n+2-2}{n+1}=\frac{2n}{n+1}$