Find the sum 2^2+4^2+6^2+8^2+ upto n terms. - Vidyadip

Question

Find the sum $2^2+4^2+6^2+8^2+\ldots \ldots$ upto $n$ terms.

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Answer

$
\begin{aligned}
& 2^2+4^2+6^2+8^2+\ldots \ldots \text { upto } \mathrm{n} \text { terms } \\
& =(2 \times 1)^2+(2 \times 2)^2+(2 \times 3)^2+(2 \times 4)^2+\ldots \ldots . \\
& =\sum_{\mathrm{r}=1}^{\mathrm{n}}(2 \mathrm{r})^2 \\
& =4 \sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2 \\
& =\frac{4 \cdot \mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6} \\
& =\frac{2 \mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{3} \\
& \text { }
\end{aligned}
$

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