Question

 In an AP given a = 3, n = 8, S_n = 192, find d.

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

Answer 1

Given that, a = 3, n = 8, S_n = 192

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

`192 = 8/2[2xx3+(8-1)d]`

192 = 4 [6 + 7d]

48 = 6 + 7d

42 = 7d

`d = 42/7`

d = 6

Answer 2

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.