if an A.P $s n = 3 n^{2} + 5 n f i n d A . P ?$

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Solution

Solution :-
Given $S_{n} = 3 n^{2} + 5 n$

$S_{1} = a_{1} = 3 (1)^{2} + 5 (1) = 3 + 5 = 8$

$S_{2} = a_{1} + a_{2} = 3 (2)^{2} + 5 (2) = 3 (4) + 10 = 22$

$\Rightarrow 8 + a_{2} = 22$

$\Rightarrow a_{2} = 14$

$S_{3} = a_{1} + a_{2} + a_{3} = 3 (3)^{2} + 5 (3) = 27 + 15 = 42$

$\Rightarrow a_{3} = S_{3} - S_{2} = 42 - 22 = 20$

$d = a_{2} - a_{1} = a_{3} - a_{2} = 14 - 8 = 20 - 14 = 6$

The terms of an A.P are $8, 14, 20 \dots \dots$

with common difference $6$

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