If the sum of first $m$ terms of an $A P$ is $3 n^{2} + 5 n$ and $m^{t h}$ terms is $164$ . Find the value of $m$ .

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Solution

Given,
$S_{n} = 3 n^{2} + 5 n$
$S_{1} = a = 3 (1)^{2} + 5 (1) = 8 = a_{1}$
$S_{2} = a + d = 3 (2)^{2} + 5 (2) = 22$
$S_{2} = a_{1} + a_{2}$
$a_{2} = S_{2} - a_{1} = 22 - 8 = 14$
$d = a_{2} - a_{1} = 14 - 8 = 6$
$a_{n} = a + (n - 1) d$
$164 = 8 + (m - 1) 6$
$164 = 8 + 6 m - 6$
$∴ m = 27$

Hence, this is the answer.

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