(Solved):   Find the sum using the formulas for the sums of powers of integers. \[ \sum_{n=1}^{7} n^{2 ...

 

Find the sum using the formulas for the sums of powers of integers. \[ \sum_{n=1}^{7} n^{2} \] Which formula for the sums of powers of integers is needed to find the sum? \[ \begin{array}{l} 1+2+3+4+\cdots+n=\frac{n(n+1)}{2} \\ 1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} \\ 1^{3}+2^{3}+3^{3}+4^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} \\ 1^{4}+2^{4}+3^{4}+4^{4}+\cdots+n^{4}= \\ \frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30} \\ 1^{5}+2^{5}+3^{5}+4^{5}+\cdots+n^{5}=\frac{n^{2}(n+1)^{2}\left(2 n^{2}+2 n-1\right)}{12} \end{array} \]