Find the sum of the series 5^{2} + 6^{2} + 7^ | vedclass

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Question

(A) The given series is $5^{2} + 6^{2} + 7^{2} + \ldots + 20^{2}$.This can be written as the difference of two sums of squares: $\sum_{k=1}^{20} k^{2}

Vedclass · Accepted Answer

$2840$

Vedclass · Answer

(A) The given series is $5^{2} + 6^{2} + 7^{2} + \ldots + 20^{2}$.This can be written as the difference of two sums of squares: $\sum_{k=1}^{20} k^{2} - \sum_{k=1}^{4} k^{2}$.Using the formula $\sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6}$:For $n=20$: $\sum_{k=1}^{20} k^{2} = \frac{20(21)(41)}{6} = 10 	imes 7 	imes 41 = 2870$.For $n=4$: $\sum_{k=1}^{4} k^{2} = \frac{4(5)(9)}{6} = 2 	imes 5 	imes 3 = 30$.Therefore,the sum is $2870 - 30 = 2840$.

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

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