Answer
12110
Work Step by Step
$S=\sum_{n=1}^{6}\left(n^{5}-n^{2}\right)$ (We have to find the given sum using formulas for the sums of powers of integers)
$S=\sum_{n=1}^{6}\left(n^{5}-n^{2}\right)$
$=\sum_{n=1}^{6} n^{5}-\sum_{n=1}^{6} n^{2}$ $=\frac{6^{2} \cdot 7^{2} \cdot\left(2 \cdot 6^{2}+2 \cdot 6-1\right)}{12}-\frac{6 \cdot 7 \cdot(2 \cdot 6+1)}{6} = 12110$
(We compute the sum, using the formulas:
$1^{2}+2^{2}+3^{2}+\ldots .+k^{2}=\frac{k(k+1)(2 k+1)}{6}$ $1^{5}+2^{5}+\ldots+k^{5}=\frac{k^{2}(k+1)^{2}\left(2 k^{2}+2 k-1\right)}{12}$
)
