Let sum of n terms of a series be $n (2 n - 1)$ . Find its mth terms

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Solution

Let $S_{m}$ and $S_{m - 1}$ denote the sum of the first m and (m-1) terms respectively.
$S_{m} = T_{1} + T_{2} + T_{3} + . . . . . . + T_{m - 1} + T_{m}$
$S_{m} = T_{1} + T_{2} + T_{3} + . . . . . + T_{m - 1}$
Subtracting
$S_{m} - S_{m - 1} = T_{m}$
$\Rightarrow T_{m} = (m (2 m - 1)) - (m - 1) (2 (m - 1) - 1))$
$= (2 m^{3} - m) - (2 m^{2} - 5 m + 3)$
$= 4 m - 3$

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