Question

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.

Answer 1

In the given problem, we have the first and the nth 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 nth term of the A.P (l) = 33

Sum of all the terms  S_n = 123

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 = (n/2)( 8 + 33)`

        `123 = (n/2) (41) `

`((123)(2))/41 = n`

                `n = 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

`(33-8)/5 = d`

Further, solving for d,

`d =25/5`

 d = 5

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