Question

Find m if the following equation holds true $1.2+2.3+3.4+..+n(n+1)=\frac{1}{m}n(n+1)(n+2)$

Answer 1

$n$th term of the given series is, $a_n=n(n+1)$

$a_n=n^2+n$

Sum of $n$ terms is $S_n={\sum }_{k=1}^n{k}^2+{\sum }_{k=1}^{n}k$

$=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}$

$=\frac{n(n+1)}{2}[\frac{2n+1}{3}+1]$

$=\frac{n(n+1)}{2}\cdot \frac{2n+4}{3}$

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

$\Rightarrow \frac{1}{m}=\frac{1}{3}$

$m=3$