Question

Find the sum to $n$ terms of the series $1\times 2+2\times 3+3\times 4+......................$ to n terms

Answer 1

$1\times 2+2\times 3+\cdots nterms$

n th term is $n(n+1)=n^2+n$

$\sum n^2+n\sum n^2+\sum n\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\frac{n(n+1)}{2}(1+\frac{2n+1}{3})\frac{n(n+1)(2n+4)}{6}$