Given: pth term is q and (p + q)th term is 0.

To prove: qth term is p.

p

th term is given by

q = a + (p – 1) × d……equation1

(p + q)th term is given by

0 = a + (p + q – 1) × d

0 = a + (p – 1) × d + q × d

Using equation1

0 = q + q × d

d = – 1

Put in equation1 we get

a = q + p – 1

q

th term is

⟹ q + p – 1 + (q – 1) × ( – 1)

⟹ p

Hence proved.

In an AP, the pth term is q and (p + q)th term is 0. Show that its qth term is p.

In an AP, the pth term is q and (p + q)th term is 0. Show that its qth term is p.