In an A.P. if pth term is q and qth term is p. where p ≠ q. Find the mth term.

Solution:

Let a be the first term and d the common difference. The n-th term of an A.P. is T_n = a + (n - 1)d.

• Given T_p = q, so a + (p - 1)d = q.

• Given T_q = p, so a + (q - 1)d = p.

• Subtract the two equations: (p - q)d = q - p = -(p - q). Since p ≠ q, divide by (p - q) to get d = -1.

• Substitute d = -1 into a + (p - 1)d = q to find a: a + (p - 1)(-1) = q so a = p + q - 1.

• Now the mth term is T_m = a + (m - 1)d = (p + q - 1) + (m - 1)(-1) = p + q - m.

Therefore the mth term equals p + q - m.

Quick check: for p = 1, q = 2, m = 3 we get T_3 = 1 + 2 - 3 = 0, which matches a direct construction of the A.P. with those positions.