Question

If mth term of an A.P. is n and nth term is m, then write its pth term.

Answer 1

Given:

$$\begin{gathered} a_m=n \\ \Rightarrow a+\left(m-1\right)d=n....\left(1\right) \\ {a}_n=m \\ \Rightarrow a+(n-1)d=m....\left(2\right) \end{gathered}$$

Solving equations $\left(1\right)\mathrm{and}\left(2\right)$, we get:
d = $-1$
a = n+m$-1$

pth term:
$$\begin{gathered} a_p=a+\left(p-1\right)d \\ =n+m-1+\left(p-1\right)\left(-1\right) \\ =n+m-p \end{gathered}$$

Hence, the pth term is n + m $-$p.