Sum of first n terms of an arithmetic sequence is $3 n^{2} + n$ . Find the first term and common difference of this sequence.

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Solution

Sum of first n terms of an arithmetic sequence is $3 n^{2} + n$ .
Let $a_{1}$ denotes first term of the sequence,
$a_{2}$ denotes second term of the sequence,
d denotes common difference of the sequence,
$S_{n}$ denotes sum of first n terms of the sequence.
$S_{n} = 3 n^{2} + n$
$S_{1} = a_{1} = 3 (1)^{2} + 1 = 4$
$S_{2} = a_{1} + a_{2} = 3 (2)^{2} + 2 = 14$
$∴ a_{1} + a_{2} = 14$
$∴ a_{1} + a_{1} + d = 14$
$∴ 2 a_{1} + d = 14$
$∴ 2 (4) + d = 14$
$∴ d = 6$
First term of the sequence is 4 and common difference is 6.

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