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