Question

In an $A.P.$ if $m^{th}$ term is $n$ and $n^{th}$ term is $m$, where $m\ne n,$ find the $p^{th}$ term.

Answer 1

Finding $p^{th}$ term.
Given, $a_m=n$ and $a_n=m$, where $m\ne n$
Let $d=$ common difference and $a=$ first term
We have,
$a_m=a+(m-1)d=n\cdots \left(i\right)$
$a_n=a+(n-1)d=m\cdots \left(ii\right)$

Subtracting equation $\left(i\right)$ and $\left(ii\right)$
$\Rightarrow (m-1)d-(n-1)d=n-m$
$\Rightarrow md-d-nd+d=n-m$
$\Rightarrow (m-n)d=n-m$
$\Rightarrow d=-1$ $($Substitute in equation $\left(i\right))$
$a+(m-1)(-1)=n$
$\Rightarrow a=n+(m-1)=n+m-1$

Now, $p^{th}$ term
$a_p=a+(p-1)d$
$=(n+m-1)+(p-1)(-1)$
$=n+m-p$
Hence, the $p^{th}$ term is $n+m-p$.