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:

• T_n ​ 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:

• S_n ​ 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.