The sum of the first n even natural numbers i | vedclass

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(A) The first $n$ even natural numbers are $2, 4, 6, \dots, 2n$.This forms an Arithmetic Progression $(AP)$ where the first term $a = 2$,the common di

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$n^{2}+n$

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(A) The first $n$ even natural numbers are $2, 4, 6, \dots, 2n$.This forms an Arithmetic Progression $(AP)$ where the first term $a = 2$,the common difference $d = 2$,and the number of terms is $n$.The sum of an $AP$ is given by the formula $S_n = \frac{n}{2}[2a + (n-1)d]$.Substituting the values: $S_n = \frac{n}{2}[2(2) + (n-1)2]$.$S_n = \frac{n}{2}[4 + 2n - 2]$.$S_n = \frac{n}{2}[2n + 2]$.$S_n = \frac{n}{2} 	imes 2(n + 1)$.$S_n = n(n + 1) = n^2 + n$.

The sum of the first $n$ even natural numbers is........

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