The sum of first $n$ terms of an A.P. is $3 n^{2} - 2 n,$ its $19^{t h}$ term is

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Solution

$S_{n} = 3 n^{2} - 2 n$

$S_{1} = 3 {(1)}^{2} - 2 (1)$
$= 3 - 2 = 1 = a_{1}$

$S_{2} = 3 {(2)}^{2} - 2 (2)$
$= 8$

$a_{2} = S_{2} - a_{1}$
$= 8 - 1 = 7$

$d = a_{2} - a_{1}$
$= 7 - 1 = 6$

$a_{n} = a + (n - 1) d$

$a_{19} = 1 + (19 - 1) \times 6$
$= 1 + 18 \times 6$
$= 1 + 108$
$= 109$

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