Question

The pth and qth terms of an A.P. are q and p, respectively. Prove that its (p + q)th terms will be zero.

Answer 1

Given:

The pth term of an A.P. is q and the qth term is p.

To prove:

The (p + q)th term is 0.

Using the formula a_n = a + (n − 1)d

a + (p − 1)d = q  ...(1)

a + (q − 1)d = p  ...(2)

Subtract (2) from (1):

(p − q)d = q − p

⇒ d = −1

Substitute d = − 1in (1)

a + (p − 1)d = q

a + (p − 1)(−1) = q

a − (p − 1) = q

⇒ a = p + q − 1

Now, the (p + q)th term:

`a_(p + q) = a + (p + q − 1)d`

= (p + q − 1) + (p + q − 1)(− 1)

= 0

∴ The (p + q)th term of A.P. is 0.