Question

Sum to $n$ terms of the series $\frac{4}{\mathrm{1.2.3}}+\frac{5}{\mathrm{2.3.4}}+\frac{6}{\mathrm{3.4.5.}}+...$

Answer 1

Let $T_r=\frac{r+3}{r(r+1)(r+2)}=\frac{1}{(r+1)(r+2)}+\frac{3}{r(r+1)(r+2)}$
$=[\frac{1}{r+1}-\frac{1}{r+2}]+\frac{3}{2}[\frac{1}{r(r+1)}-\frac{1}{(r+1)(r+2)}]$
$\therefore$ $S=[\frac{1}{2}-\frac{1}{n+2}]+\frac{3}{2}[\frac{1}{2}-\frac{1}{(n+1)(n+2)}]$
$=\frac{5}{4}-\frac{1}{n+2}[1+\frac{3}{2(n+1)}]$
$=\frac{5}{4}-\frac{1}{2(n+1)(n+2)}[2n+5]$