Question

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

Answer 1

Given,

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

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

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

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

$\therefore d=-1$

From (1)

$a=m+n-1$

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

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

$=m+n-p$

Hence, this is the answer.