Question

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

Answer 1

Formula,

$a_n=a+(n-1)d$

Given,

$a_m=a+(m-1)d=n$........(1)

$a_n=a+(n-1)d=m$......(2)

$n-m=a+(m-1)d-[a+(n-1\left)d\right]$

$n-m=d(m-n)$

$\therefore d=-1$

substituting the value d in eqn (1), we get,

$a=m+n-1$

Therefore the pth term is given by,

$a_p=a+(p-1)d$

$=m+n-1+(p-1)(-1)$

$=m+n-1-p+1$

$=m+n-p$