If the sum of the first $n$ terms of an A.P is given $Sn$= $2n^2+5n$. Find the $n$th term of the A.P?

Question

If the sum of the first $n$ terms of an A.P is given $Sn$= $2n^2+5n$. Find the $n$th term of the A.P ?

Answer 1

$a_n = nd+a_0$

$S_n = $$d\sum_\limits{i=1}^n k + na_1\\ d(\frac12)(n^2 + n) + na_0\\\frac d2 = 2, \frac d2 + a_0 = 5\\ d = 4, a_0 = 3\\ a_n = 4n + 3$

Alternate:

$S_n = S_{n-1} + a_n\\ 2n^2 + 5n = 2(n-1)^2 + 5(n-1) + a_n\\ 2n^2 + 5n = 2n^2 - 4n + 2 + 5n - 5 + a_n\\ 4n+3 = a_n$