In an A.P., the first term is 22, nth term is $-11$ and the sum to first n terms is 66. Find n and d, the common difference.

Given:

In an A.P., the first term is 22, nth term is $-11$ and the sum to first n terms is 66. 

To do:

We have to find the value of $n$ and $d$, the common difference.

Solution:

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

First term $a=22$

nth term $l=a+(n-1)d$

$-11=22+(n-1)d$

$(n-1)d=-11-22$

$(n-1)d=-33$.....(i)

Sum of n terms $S_{n} =66$

We know that,

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

$\Rightarrow 66=\frac{n}{2}[2(22)+(n-1)d]$

$\Rightarrow 66=\frac{n}{2}[44+(-33)]$         (From (i))

$\Rightarrow 66(2)=11n$

$\Rightarrow n=6(2)$

$\Rightarrow n=12$

This implies,

$(12-1)d=-33$

$11d=-33$

$d=-3$

The value of $n$ is $12$ and the value of $d$ is $-3$.