If the sum of n terms of an A.P is (pn + qn²), where p and q are constants, find the common difference

Solution:

We know that S_n = n/2 [2a + (n - 1) d]

According to the question,

n/2 [2a + (n - 1) d] = pn + qn²

n/2 [2a + nd - d] = pn + qn²

an + 1/2 dn² - 1 dn = pn + qn²

(a - 1/2d)n + 1/2dn² = pn + qn²

Comparing the coefficients of n² on both sides, we obtain

⇒ 1/2 d = q

⇒ d = 2q

Thus, the common difference is 2q

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.2 Question 8

If the sum of n terms of an A.P is (pn + qn²), where p and q are constants, find the common difference

Summary:

The sum of an A.P is (pn + qn²) and we also knew that S_n = n/2 [2a + (n - 1) d] therefore comparing those two the common difference is 2q