What is the average of the squares of the fir | vedclass

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(C) The sum of the squares of the first $n$ natural numbers is given by the formula: $S_n = \frac{n(n+1)(2n+1)}{6}$.For $n = 19$,the sum is $S_{19} =

Vedclass · Accepted Answer

$130$

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(C) The sum of the squares of the first $n$ natural numbers is given by the formula: $S_n = \frac{n(n+1)(2n+1)}{6}$.For $n = 19$,the sum is $S_{19} = \frac{19(19+1)(2 	imes 19 + 1)}{6} = \frac{19 	imes 20 	imes 39}{6}$.The average is defined as the sum divided by the total number of terms $(n)$:Average $= \frac{S_n}{n} = \frac{n(n+1)(2n+1)}{6n} = \frac{(n+1)(2n+1)}{6}$.Substituting $n = 19$:Average $= \frac{(19+1)(2 	imes 19 + 1)}{6} = \frac{20 	imes 39}{6} = \frac{780}{6} = 130$.

What is the average of the squares of the first $19$ natural numbers?

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