If the p^th term of an A. P. is q and q^th term is p, prove that its n^th term is (p + q – n).

Given: a_p = q and a_q = p

To Prove: a_n = (p + q – n)

Proof:

We know that,

a_n = a + (n – 1)d

Putting n = p

a_p = a + (p – 1)d

⇒ q = a + (p – 1)d

⇒ q – (p – 1)d = a …(i)

Similarly,

a_q = a + (q – 1)d

⇒ p = a + (q – 1)d

⇒ p – (q – 1)d = a …(ii)

From eq. (i) and (ii), we get

q – (p – 1)d = p – (q – 1)d

⇒ q – pd + d = p – qd + d

⇒ qd – pd = p – q

⇒ d(q – p) = -(q – p)

⇒ d = -1

Putting the value of d in eq. (i), we get

q – (p – 1)(-1) = a

⇒ q + p – 1 = a

Now, we have to find n^th term

We know that,

a_n = a + (n – 1)d

Putting a = (q + p – 1) and d = -1

a_n = (p + q – 1) + (n – 1)(-1)

⇒ a_n = p + q – 1 – n + 1

⇒ a_n = p + q – n

Hence Proved