\(5^{2}+6^{2}+7^{2}+\ldots \ldots \ldots \ldots \ldots+20^{2}=\)

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 \((16+4)^{2}=20^{2}\)
\(\therefore S_{10} =\frac{16(16+1)(2 \times 16+1)}{6}+\frac{8 \times 16 \times(16+1)}{2}+16 \times 16 \)
\( =\frac{(16)(17)(33)}{6}+\frac{(8) \times 16 \times(16+1)}{2}+16 \times 16 \)
\( =\frac{(16)(17)(33)}{6}+\frac{(8)(16)(17)}{2}+256 \)
\( =1496+1088+256 \)
\( =2840 \)
\( \therefore 5^{2} +6^{2}+7^{2}+\ldots \ldots+20^{2}=2840\)