If the sum of first n term is (3n^2 + 5n), find its common differ… - Vidyadip

Question

If the sum of first n term is $(3n^2 + 5n)$, find its common difference.

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Answer

Let $S_n$ denotes the sum of first n terms of the AP.
$\therefore$ $S_n = 3n^2 + 5n$
$\Rightarrow S_{n - 1} = 3(n - 1)^2 + 5(n - 1)$
$= 3(n^2 - 2n + 1) + 5(n - 1)$
$= 3n^2 - n - 2$
Now,
$n^{th}$ term of the AP, $a_n = S_n - S_{n - 1}$
$= (3n^2 + 5n) - (3n^2 - n - 2)$
$= 6n + 2$
Let d be the common difference of the AP.
$\therefore$ $d = a_n - a_{n -1}$
$= (6n + 2) - [6(n - 1) + 2]$
$= 6n + 2 - 6(n - 1) - 2$
$= 6$
Hence, the common difference of the AP is $6$.

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