Question

If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero.

Answer 1

In the given AP, let the first be a and the common difference be d.

Then, T_n = a + (n – 1)d 

⇒ Tp = a + (p – 1)d = q   ...(i)

⇒ T_q = a + (q – 1)d = p   ...(ii)

On subtracting (i) from (ii), we get:

(q – p)d = (p – q) 

⇒ d = –1

Putting d = –1 in (i), we get:

a = (p + q – 1)

Thus, a = (p + q – 1) and d = –1 

Now, T_{p + q} = a + (p + q – 1)d

= (p + q – 1) + (p + q – 1)(–1)

= (p + q – 1) – (p + q – 1) = 0

Hence, the (p + q)^th term is 0 (zero).