Question

The sum of first n terms of an A.P is $S_n=5n^2+2n$. Find the ${18}^{th}$ term of the A.P

Answer 1

Given

$S_n=5n^2+2n$ ___________ (1)

then $S_{n-1}=5(n-1{)}^2+2(n-1)$

$S_{n-1}=5n^2+5-10n+2n-2$

$S_{n-1}=5n^2-8n+3$ _________ (2)

We know $S_n-S_{n-1}=a_n$

then $a_n=5n^2+2n-5n^2+8n-3=10n-3$ _________ (3)

and $a_n=a_{n-1}=d$

$10n-3-10n+10+3=d$

[d=10]

$\therefore$ common difference of AP is 10.

$a_n=a+(n-1)10\Rightarrow 10n-3=a+10n-10$

[a=7]

now ${18}^{th}$ term of AP having a=7 & d=10

$a_{18}=7+17d=7+170=177$

$\therefore {18}^{th}$ term is 177.