Arithmetic progression

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Question

The sum of $n$ terms of an $A . P .$ is $3 n^{2} + 5$ The number of term which equals 159, is

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Solution

Let $S_{n}$ denote the sum of $n$ terms. Then,
$S_{n} = 3 n^{2} + 5$
Now,
$\begin{array}{l} a_{n} = S_{n} - S_{n - 1} \\ a_{n} = (3 n^{2} + 5) - (3 (n - 1)^{2} + 5) = 6 n - 3 \\ a_{n} = 159 \Rightarrow 6 n - 3 = 159 \Rightarrow 6 n = 162 \Rightarrow n = 27 . \end{array}$

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