Suppose that m and n are fixed numbers such that the mth term of an HP is equal to n and the nth term is equal to m, (m ≠ n). Then the (m + n)th term is:

Given: The sequence is in H.P. with \(a_m = n\) and \(a_n = m\) \((m \ne n)\).

In an H.P., reciprocals of terms form an A.P. So, \(\tfrac{1}{a_m} = \tfrac{1}{n}\) and \(\tfrac{1}{a_n} = \tfrac{1}{m}\).

This A.P. leads to common difference \(d = \tfrac{1}{mn}\) and first term \(A = \tfrac{1}{mn}\).

Thus, the reciprocal of the \((m+n)\)th term is:

\(b_{m+n} = \tfrac{m+n}{mn}\)

Hence, the \((m+n)\)th term of H.P. is:

\(a_{m+n} = \dfrac{mn}{m+n}\)