Answer
$\sum\limits_{i =1}^{n}i(i+1)(i+2)=\frac{1}{4}n(n+1)[n^{2}+5n+6]$
Work Step by Step
Find the value of the sum $\sum\limits_{i =1}^{n}i(i+1)(i+2)$
After expanding the terms, we have
$\sum\limits_{i =1}^{n}i(i+1)(i+2)=\sum\limits_{i =1}^{n}(i^{3}+3i^{2}+2i)$
$=\sum\limits_{i =1}^{n}i^{3}+3\sum\limits_{i =1}^{n}i^{2}+2\sum\limits_{i =1}^{n}i$
$=[\frac{n(n+1)}{2}]^{2}+3[\frac{n(n+1)(2n+1)}{6}]+2[\frac{n(n+1)}{2}]$
$=\frac{1}{4}n^{2}(n+1)^{2}+\frac{1}{2}n(n+1)(2n+1)+n(n+1)$
$=\frac{1}{4}n(n+1)[n^{2}+n+4n+2+4]$
$=\frac{1}{4}n(n+1)[n^{2}+5n+6]$
Hence, $\sum\limits_{i =1}^{n}i(i+1)(i+2)=\frac{1}{4}n(n+1)[n^{2}+5n+6]=\frac{1}{4}n(n+1)(n+2)(n+3)$
