# If the sum of n terms of an A.P is (pn + qn^{2}), 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 + qn^{2}

n/2 [2a + nd - d] = pn + qn^{2}

an + 1/2 dn^{2} - 1 dn = pn + qn^{2}

(a - 1/2d)n + 1/2dn^{2 }= pn + qn^{2}

Comparing the coefficients of n^{2} 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 + qn^{2}), where p and q are constants, find the common difference

**Summary:**

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

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