The sum of $r$ terms of an AP is $2 r^{2} + 3 r .$ The $n^{th}$ term is ___.

$2 n - 1$

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$4 n + 1$

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$3 n - 1$

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$3 n + 1$

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Solution

The correct option is B
$4 n + 1$

Given, $S_{r} = (2 r^{2} + 3 r),$ the sum upto $r$ terms of an AP.
We know that the $n^{th}$ term of AP is
$T_{n} = S_{n} - S_{n - 1} .$
$\Rightarrow S_{n} = 2 n^{2} + 3 n$ and
$S_{n - 1} = 2 {(n - 1)}^{2} + 3 (n - 1) .$

$∴ T_{n} = S_{n} - S_{n - 1}$
$= 2 n^{2} + 3 n - [2 (n^{2} - 2 n + 1) + 3 n - 3]$
$= 2 n^{2} + 3 n - 2 n^{2} + 4 n - 2 - 3 n + 3 = 4 n + 1$

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