Question

The $m^{th}$ term of ap is $n$ and its $n^{th}$ term is $m$, then write its $p^{th}$ term.

Answer 1

Find the $p^{th}$ term.

Given,
$$\begin{gathered} a_m=a+(m-1)d=n\dots \text{(i)} \\ {a}_n=a+(n-1)d=m\dots \text{(ii)} \end{gathered}$$
solve both the given equations
$d=-1$
and
$$\begin{gathered} a+(1-m)=n \\ \Rightarrow a=m+n-1 \\ \Rightarrow ap=a+(p-1)d=m+n-1+1-p \\ \Rightarrow a_p=m+n-p \end{gathered}$$

Hence, the $p^{th}$ term is $a_p=m+n-p$.