If the sum of $n$ terms of an $A . P$ is $S_{n} = 3 n^{2} + 5 n$ Write its common difference.

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Solution

Given that,

Sum of A.P. is

$S_{n} = 3 n^{2} + 5 n$

$\Rightarrow S_{n} = n (3 n + 5)$

$\Rightarrow S_{n} = \frac{2 n}{2} (3 n + 5)$

$\Rightarrow S_{n} = \frac{n}{2} (6 n + 10)$

$\Rightarrow S_{n} = \frac{n}{2} (10 + 6 n)$

$\Rightarrow S_{n} = \frac{n}{2} (10 + 6 n + 6 - 6) (a d d i n g a n d s u b t r a c t i n g 6)$

$\Rightarrow S_{n} = \frac{n}{2} (16 + 6 n - 6)$

$\Rightarrow S_{n} = \frac{n}{2} [16 + 6 (n - 1)]$

$\Rightarrow S_{n} = \frac{n}{2} [2 \times 8 + (n - 1) 6]$

Comparing that $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

Then, first term $a = 8$ and common difference $d = 6$

Hence, this is the answer.

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