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Sum of n Terms in AP

If the sum of...

Question

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

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Solution

Given, $s_{n} = 3 n^{2} + 5 n, a_{m} = 164$
We know,
$a_{n} = s_{n} - s_{n - 1}$
$= 3 n^{2} + 5 n - 3 (n - 1)^{2} - 5 (n - 1)$
$= 3 n^{2} + 5 n - 3 n^{2} + 6 n - 3 - 5 n + 5$
$∴ a_{n} = 6 n + 2$
$\Rightarrow a_{m} = 6 m + 2$
$\Rightarrow 164 = 6 m + 2$
$\Rightarrow 6 m = 162$
$\Rightarrow m = 27$

Final answer : Hence, the value of $m = 27$

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