If the sum of first $m$ terms of an A.P. is $2 m^{2} + 3 m$ , then what is its second term?

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Solution

$s m = 2 m^{2} + 3 m$
$s m - 1 = 2 {(m - 1)}^{2} + 3 (m - 1)$
$= 2 (m^{2} + 1 - 2 m) + 3 m - 3$
$= 2 m^{2} + 2 - 4 m + 3 m - 3$
$= 2 m^{2} - m - 1$
$∴ s m - s m - 1 = 2 m^{2} - + 3 m - (2 m^{2} - m - 1) = 2 m^{2} + 3 m - 2 m^{2} + m + 1$
$= 4 m + 1$
General term $= a m = 4 m + 1$
Second term $= a_{2} = 4 (2) + 1 = 8 + 1 = 9$

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