Question

If the sum of the first n terms of an A.P. is \( 4n^2 + 2n \), then the common difference of the A.P. is:

• A. 6
• B. 14
• C. 8
• D. 4

Hint:

Use the difference of sums formula to find the common difference when the sum of terms is given.

Answer 1

The Correct Option is D

The sum of the first n terms of an A.P. is given by: \[ S_n = \frac{n}{2} \cdot (2a_1 + (n - 1) d). \] We are given that: \[ S_n = 4n^2 + 2n. \] To find the common difference \( d \), we take the difference between \( S_n \) and \( S_{n-1} \), i.e., \( S_n - S_{n-1} \): \[ S_n - S_{n-1} = a_n. \] Differentiating \( S_n \) with respect to \( n \) to find the common difference: \[ \frac{d}{dn}(4n^2 + 2n) = 8n + 2. \] Thus, the common difference \( d \) is \( 8n + 2 \). Therefore, the correct answer is \( \boxed{4} \).