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

Given:

In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123.

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 $a=8$

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

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

$(n-1)d=33-8$

$(n-1)d=25$.....(i)

Sum of n terms $S_{n} =123$

We know that,

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

$\Rightarrow 123=\frac{n}{2}[2(8)+(n-1)d]$

$\Rightarrow 123=\frac{n}{2}(16+25)$         (From (i))

$\Rightarrow 123(2)=41n$

$\Rightarrow n=3(2)$

$\Rightarrow n=6$

This implies,

$(6-1)d=25$

$5d=25$

$d=5$

The value of $n$ is $6$ and the value of $d$ is $5$.