**Q) ** Find the common difference of an A.P. whose first term is 8, the last term is 65 and the sum of all its terms is 730.

**Ans: **In the AP question, we are given, a = 8 Last term T_{n }= 65, Sum of AP S_{n }= 730; We need to find value of common difference d.

We know that the n^{th} term of an AP is given by: T_{n } = a + (n – 1) d

here:

- $Tn $ is the n
^{th}term - $a$ is the first term,
- $d$ is the common difference,
- $n$ is the number of terms.

In this problem, we are given the sum formula as T_{n }$= 65$

∴ 8 + (n – 1) x d = 65

∴ (n – 1) d = 57 ……………….. (i)

Next, we know that, the formula for the sum of the first n terms of an arithmetic progression (AP) is given by:

S_{n }_{ }= [2a + (n-1) d]

here:

- $Sn $ is the sum of the first n terms,
- $a$ is the first term,
- $d$ is the common difference,
- $n$ is the number of terms.

In this problem, we are given the sum formula as S_{n }$= 730$.

∴ [2 x 8 + (n-1) d] = 730^{ }

∴ n [16 + (n – 1) d] = 1460 ………… (ii)

By substituting value from equation (i) in equation (ii), we get:

n (16 + 57) = 1460

∴ n = 20

by putting n = 20 in equation (i), we get:

(n -1) d = 57

∴ (20 -1) d = 57

∴ d = 3

**Therefore, the value of common difference is 3.**