#### Question

Let there be an A.P. with the first term 'a', common difference 'd'. If a_{n} a denotes in n^{th} term and S_{n} the sum of first n terms, find.

d, if a = 3, n = 8 and S_{n} = 192.

#### Solution

Here, we have an A.P. whose first term (a), the sum of first n terms (S_{n}) and the number of terms (n) are given. We need to find the common difference (d).

Here,

First term (a) = 3

Sum of *n* terms (*S*_{n}) = 192

Number of terms (*n*) = 8

So here we will find the value of *n* using the formula, `a_n = a + (a - 1)d`

So, to find the common difference of this A.P., we use the following formula for the sum of *n* terms of an A.P

`S_n = n/2 [2a + (n -1)d]`

Where; *a* = first term for the given A.P.

*d* = common difference of the given A.P.

*n *= number of terms

So, using the formula for *n* = 8, we get,

`S_8 = 8/2 [2(3) + (8 - 1)(d)]`

192 = 4[6 = (7) (d)]

192 = 24 + 28d

`28d = 192 - 24`

Further solving for d

`d = 168/28`

d= 6

Therefore, the common difference of the given A.P. is d = 6