Question

if the sum of n terms of an AP is (pn+qn^2), where p and q are constants, find the commom difference

Answer 1

Answer

2q

Explanation

Given that sum of n terms of an A.P. is given as (pn + qn²) where p and q are constants.

To find, the common difference

Here, p and q are constants but n is variable.

Since sum of first n terms = (pn + qn²)

So sum of terms when n = 1 would equal

p(1) + q(1)²

= p + q

We know that sum of first terms = first term of A.P.

This gives, first term of A.P. = p + q

Now, put n = 2

p(2) + q(2)²

= 2p + 4q

This is the sum of first and second term

⇒ first term + second term = 2p + 4q

⇒ (p + q) + second term = 2p + 4q

⇒ second term = 2p + 4q - (p + q)

⇒ second term = 2p + 4q - p - q

⇒ second term = p + 3q

We know that for an A.P., common difference = second term - first term

⇒ common difference = p + 3q - (p + q)

⇒ common difference = p + 3q - p - q = 2q

hence, the common difference = 2q.