Question

In a A.P., if the first term is $22$, the common difference is $-4$ and sum of $n$ terms is $64$: find $n$.

Answer 1

Firts term $=a=22$

Common difference $d==-4$

Sum $=S=64$

$S=\frac{n}{2}(a+a+(n-1\left)d\right)$

$S=\frac{n}{2}(2a+(n-1\left)d\right)$

$64=\frac{n}{2}(44+(n-1\left)\right(-4\left)\right)$

$128=n(44-4n+4)$

$128=n(48-4n)$

$128=48n-4n^2$

$4n^2-48n+128=0$

$n^2-12n+32=0$

$n=\frac{+12\pm \sqrt{{12}^2-4\times 32}}{2}$

$n=\frac{+12\pm \sqrt{144-128}}{2}$

$n=\frac{+12\pm \sqrt{16}}{2}$

$n=\frac{+12\pm 4}{2}$

$n=8$ or $b=4$