**Question:**

In an A.P. the first term is 8, *n*th term is 33 and the sum to first *n* terms is 123. Find *n* and *d*, the common differences.

**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*) = 8

The *n*th term of the A.P (*l*) = 33

Sum of all the terms

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

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

$123=\left(\frac{n}{2}\right)(8+33)$

$123=\left(\frac{n}{2}\right)(41)$

$\frac{(123)(2)}{41}=n$

$n=\frac{246}{41}$

$n=6$

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

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

We get,

$33=8+(6-1) d$

$33=8+(5) d$

$\frac{33-8}{5}=d$

Further, solving for *d,*

$d=\frac{25}{5}$

$d=5$

Therefore, the number of terms is $n=6$ and the common difference of the A.P. $d=5$.