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Q. The value of $\displaystyle\sum_{n=1}^{10} \displaystyle\sum_{m=1}^{10}\left(m^{2}+n^{2}\right)$ equals

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Solution:

$\displaystyle\sum_{n=1}^{10} \displaystyle\sum_{m=1}^{10}\left(m^{2}+n^{2}\right)$
$=\displaystyle\sum_{n=1}^{10}\left[\left(1^{2}+n^{2}\right)+\left(2^{2}+n^{2}\right)+\cdots+\left(10^{2}+n^{2}\right)\right]$
$=10\left[(1)^{2}+(2)^{2}+\ldots+(10)^{2}\right]+10\left[(1)^{2}+(2)^{2}+\ldots+(10)^{2}\right]$
$=\frac{20 \cdot 10 \cdot 11 \cdot 21}{6}$
$=7700$