Question

Solve: $\sum ^{n}k(k+1)(k+2)x^k$

Answer 1

${\sum }_{k=1}^{n}k(k+1)(k+2)x^k$

${\sum }_{k=1}^n(k^3+3k^2+2k)$

$S={\sum }_{k=1}^n{k}^3$$+3{\sum }_{k=1}^n{k}^2$$+2{\sum }_{k=1}^{n}k$

$S=[\frac{n(n+1)}{2}{]}^2$$+\left[\frac{n(n+1)(2n-1)}{6}\right]$$+\left[\frac{n(n+1)}{2}\right]$

$⟹S=\frac{n+(n+1)}{12}(3n^2+7n+4)$