The sum of first $n$ terms of an $A . P$ is $2 n + 3 n^{2}$ . Find the common difference.

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Solution

Sum of n terms $= S_{n} = 2 n + 3 n^{2}$

$S_{1} = 2 (1) + 3 (1)^{2} = 5$

Sum of 1st term is same as first term so $a_{1} = a = 5$

$S_{2} = 2 (2) + 3 (2)^{2} = 4 + 12 = 16$

$S_{2} = a_{1} + a_{2}$

$16 = 5 + a_{2}$

$\Rightarrow a_{2} = 11$

Common difference $d = a_{2} - a_{1} = 11 - 5 = 6$

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