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

Given:

In an A.P., the first term is 22, the common difference is $-4$ and the sum to $n$ terms is 64.

To do:

We have to find the value of $n$.

Solution:

Let the first term be $a$ and the common difference be $d$.

First term $a=22$

Common difference $d=-4$

Sum of n terms $S_{n} =64$

We know that,

Sum of the $n$ terms$ S_{n} =\frac{n}{2}( 2a+(n-1)d)$

$\Rightarrow 64=\frac{n}{2}[2(22)+(n-1)(-4)]$

$\Rightarrow 64=\frac{n}{2}(44-4n+4)$

$\Rightarrow 64=\frac{n}{2}(48-4n)$

$\Rightarrow 64=(n)(24-2n)$

$\Rightarrow 32=(n)(12-n)$

$\Rightarrow 12n-n^2=32$ 

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

$\Rightarrow n^2-8n-4n+32=0$

$\Rightarrow n(n-8)-4(n-8)=0$

$\Rightarrow (n-8)(n-4)=0$

$\Rightarrow n=8$ or $n=4$

The value of $n$ is $4$ or $8$.