# The sum of the first 9 terms of an AP is 81 and that of its first 20 terms is 400.

Question:

The sum of the first 9 terms of an AP is 81 and that of its first 20 terms is 400. Find the first term and the common difference of the AP.

Solution:

Sum of 9 term $s$ of $\mathrm{AP}=81$

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

$81=\frac{9}{2}[2 a+8 d]$

$2 a+8 d=18$                                  $\ldots \ldots(1)$

Sum of 1 st 20 terms is 400

$400=\frac{20}{2}[2 a+19 d]$

$2 a+19 d=40$                                   $\ldots \ldots(2)$

Subtract (1) from (2)

$d=2$ and $a=1$

Therefore, first term is 1 and common difference is 2