Question

what is the value of1²+2²+3²⁺+4²+5²+6²+7²+8²

Answer 1

We know that

$$\begin{gathered} \sum _{}^{}{n}^2=\frac{n(n+1)(2n+1)}{6}\mathrm{so}\mathrm{using}\mathrm{this}\mathrm{property}, \\ \mathrm{We}\mathrm{can}\mathrm{write}{1}^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2=\sum _{\mathrm{n}=1}^{\mathrm{n}=8}{\mathrm{n}}^2 \\ \mathrm{Here}\mathrm{n}=8 \\ \sum _{\mathrm{n}=1}^{\mathrm{n}=8}{\mathrm{n}}^2=\frac{\mathrm{n}(\mathrm{n}+1)(2\mathrm{n}+1)}{6} \\ =\frac{8(8+1)(2\times 8+1)}{6} \\ =\frac{8\times 9\times 17}{6}=204 \end{gathered}$$