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

#### Solution

In the given problem, we have the first and the *n*th term of an A.P. along with the sum of the *n* terms of A.P. Here, we need to find the number of terms and the common difference of the A.P.

Here,

The first term of the A.P (*a*) = 22

The *n*th term of the A.P (*l*) = −11

Sum of all the terms S_{n} = 66

Let the common difference of the A.P. be *d*.

So, let us first find the number of the terms (*n*) using the formula,

`66 = (n/2) [ 22 + (-11)]`

`66 = (n/2 ) (22 - 11)`

( 66)(2) = (n)(11)

Further, solving for *n*

`n =( (66)(2))/11`

n = (6) (2)

n = 12

Now, to find the common difference of the A.P. we use the following formula,

l = a + ( n-1) d

We get,

-11 = 22 + ( 12 - 1) d

- 11 = 22 + ( 11) d

`(-11 - 22) /11 = d`

Further, solving for *d,*

`d =( -33)/11`

d = -3

Therefore, the number of terms is n = 12 and the common difference of the A.P.d = -3 .