In an A.P. if mth term is n and the nth term is m, where m ≠ n, find…

The correct answer is Option (4) → \(n+m-p\)

The \(k\)th term of an AP with first term \(a\) and common difference \(d\) is \(a+(k-1)d\). The two given terms give two equations in \(a\) and \(d\); once both are known, the \(p\)th term follows.

The two terms given:

• \(a+(m-1)d=n\)        (1)

• \(a+(n-1)d=m\)        (2)

Finding \(d\) (subtracting (2) from (1), \(a\) cancels):

• \((m-n)d=n-m\)

• \(d=-1\) (dividing by \(m-n\) is allowed because \(m\ne n\))

Finding \(a\) (from (1)):

• \(a-(m-1)=n\), so \(a=n+m-1\)

The \(p\)th term:

• \(a+(p-1)d=(n+m-1)-(p-1)=n+m-p\)