Question:
The sum, $\sum_{n=1}^{7} \frac{n(n+1)(2 n+1)}{4}$ is equal to__________.
Solution:
$\left[\sum_{n=1}^{7} \frac{n(n+1)(2 n+1)}{4}\right] \frac{1}{4}\left[\sum_{n=1}^{7}\left(2 n^{3}+3 n^{2}+n\right)\right]$
$=\frac{1}{4}\left(2\left(\frac{7.8}{2}\right)^{2}+3\left(\frac{7.8 .15}{6}\right)+\frac{7.8}{2}\right)$
$\Rightarrow \quad \frac{1}{4}[2 \times 49 \times 16+28 \times 15+28]$
$=\frac{1}{4}[1568+420+28]=504$
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