Question:
Find the sum to $n$ terms of the series $1 \times 2+2 \times 3+3 \times 4+4 \times 5+\ldots$
Solution:
The given series is $1 \times 2+2 \times 3+3 \times 4+4 \times 5+\ldots$
$n^{\text {th }}$ term, $a_{n}=n(n+1)$
$\therefore S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n} k(k+1)$
$=\sum_{k=1}^{n} k^{2}+\sum_{k=1}^{n} k$
$=\frac{n(n+1)(2 n+1)}{6}+\frac{n(n+1)}{2}$
$=\frac{n(n+1)}{2}\left(\frac{2 n+1}{3}+1\right)$
$=\frac{n(n+1)}{2}\left(\frac{2 n+4}{3}\right)$
$=\frac{n(n+1)(n+2)}{3}$