Question

Given, $S_n=90,a=2,d=8$, find $n$.

Answer 1

We have,

$a=2$

$d=8$

$S_n=90$

$n=?$

Since,

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

Thus,

$90=\frac{n}{2}[2\times 2+(n-1)8]$

$180=n[4+8n-8]$

$180=n[8n-4]$

$180=8n^2-4n$

$2n^2-n-45=0$

$n=\frac{1\pm \sqrt{1-4\times 2\times -45}}{2\times 2}$

$n=\frac{1\pm \sqrt{1+360}}{4}$

$n=\frac{1\pm \sqrt{361}}{4}$

$n=\frac{1\pm 19}{4}$

$n=5,\frac{-9}{2}$

We cannot consider $n=-\frac{9}{2}$.

Hence, $n=5$.