Let there be an A.P. with first term ‘$a$’, common difference '$d$'. If $a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms, find.$S_{22}$, if $d = 22$ and $a_{22} = 149$.

Given:

In an A.P., first term $=a$ and common difference $=d$.

$a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms.

To do:

We have to find $S_{22}$, if $d = 22$ and $a_{22} = 149$.

Solution:

We know that,

$\mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d]$

$n$th term $a_n=a+(n-1)d$

This implies,

$a_{22}=a+(22-1)22$

$149=a+21(22)$

$149=a+462$

$a=149-462$

$a=-313$

$S_{22}=\frac{22}{2}[2 \times (-313)+(22-1) \times 22]$

$=11[-626+21 \times 22]$

$=11(-626+462)$

$=11 \times (-164)$

$=-1804$

Therefore, $S_{22}=-1804$.