Square of any even natural number i.e., $(2 n)^{2}$ is equal to sum of n terms of a certain series of integers in A.P.Find the Series

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Solution

$S_{n} = (2 n)^{2}$
$∴ T_{n} = S_{n} - S_{n - 1} = (2 n)^{2} - (2 n - 2)^{2}$
or $T_{n} = (4 n - 2) 2 = 8 n - 4$
On putting $n = 1, 2, 3$ ,_
we get the A.P. as $4, 12, 20, 28$ ,_.

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