The sum of the first n terms of an A.P. is $3 n^{2} + 6 n$ . Find the nth term of this A.P.

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Solution

$s_{n} = 3 n^{2} + 6 n$
$s_{n - 1} = 3 (n - 1)^{2} + 6 (n - 1)$
$s_{n - 1} = 3 (n^{2} - 2 n + 1) + 6 n - 6$
$s_{n - 1} = 3 (n^{2} - 2 n + 1) + 6 n - 6$
$s_{n - 1} = 3 n^{2} - 6 n + 3 + 6 n - 6$
$s_{n - 1} = 3 n^{2} - 3$
$t_{n} = s_{n} - s_{n - 1}$
$t_{n} = 3 n^{2} + 6 n - 3 n^{2} + 3$
$t_{n} = 6 n + 3$

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