In an A.P., if the \(m^{\text{th}}\) term is n and the \(n^{\text{th}}\) term is m, where \(m \neq n\), then the \(p^{\text{th}}\) term is __________.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(n+m-p\)
The kth term of an A.P. with first term a and common difference d is \(a+(k-1)d\). The two given terms give two equations in a and d:
\(a+(m-1)d=n\) (1)
\(a+(n-1)d=m\) (2)
Finding d and a (subtracting (2) from (1), a cancels):
\((m-n)d=n-m\), so \(d=-1\), since \(m\neq n\)
From (1): \(a-(m-1)=n\), so \(a=n+m-1\)
The pth term:
\(a+(p-1)d=(n+m-1)-(p-1)=n+m-p\)