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# Find the sum to n terms of each of the series in 5^2 + 6^2 + 7^2 + ... + 20^2

5.   Find the sum to n terms of each of the series in $5 ^ 2 + 6 ^ 2 + 7 ^ 2 + ....+ 2 0 ^2$

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series =    $5 ^ 2 + 6 ^ 2 + 7 ^ 2 + ....+ 2 0 ^2$

n th term  = $(n+4)^2=n^2+8n+16=a_n$

$S_n=\sum _{k=1}^{n} a_k=\sum _{k=1}^{n} (k+4)^2$

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

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

16th term is $(16+4)^2=20^2$

$S_1_6=\frac{16(16+1)(2(16)+1)}{6}+\frac{8.(16)(16+1)}{2}+16(16)$

$S_1_6=\frac{16(17)(33)}{6}+\frac{8.(16)(17)}{2}+16(16)$

$S_1_6=1496+1088+256$

$S_1_6=2840$

Hence, the sum of the series $5 ^ 2 + 6 ^ 2 + 7 ^ 2 + ....+ 2 0 ^2$  is 2840.

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