Let there be an A.P. with the first term 'a', common difference 'd'. If an a denotes in nth term and Sn the sum of first n terms, find.
d, if a = 3, n = 8 and Sn = 192.
Here, we have an A.P. whose first term (a), the sum of first n terms (Sn) and the number of terms (n) are given. We need to find the common difference (d).
First term (a) = 3
Sum of n terms (Sn) = 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`
Therefore, the common difference of the given A.P. is d = 6