Find the sum to n terms of the series


Find the sum to n terms of the series $5^{2}+6^{2}+7^{2}+\ldots+20^{2}$


The given series is $5^{2}+6^{2}+7^{2}+\ldots+20^{2}$

$n^{\text {th }}$ term, $a_{n}=(n+4)^{2}=n^{2}+8 n+16$

$\therefore S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n}\left(k^{2}+8 k+16\right)$

$=\sum_{k=1}^{n} k^{2}+8 \sum_{k=1}^{n} k+\sum_{k=1}^{n} 16$

$=\frac{n(n+1)(2 n+1)}{6}+\frac{8 n(n+1)}{2}+16 n$

$16^{\text {th }}$ term is $(16+4)^{2}=20^{2} 2$

$\therefore S_{16}=\frac{16(16+1)(2 \times 16+1)}{6}+\frac{8 \times 16 \times(16+1)}{2}+16 \times 16$

$=\frac{(16)(17)(33)}{6}+\frac{(8) \times 16 \times(16+1)}{2}+16 \times 16$




$\therefore 5^{2}+6^{2}+7^{2}+\ldots \ldots+20^{2}=2840$


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