Question

The $m^{th}$ terms of an A.P. is $n$ and its $n^{th}$ term is $m$. Prove that its $p^{th}$ therm is $m+n-p$. Also show that its $(m+n{)}^{th}$ term is zero.

Answer 1

let first term of AP = a and common difference = d

kth term $t_k=a+(k-1)d$

$then{m}^{th}term=n$ .... Given

$ora+(m-1)d=n.....\to \left(1\right)$

$and{n}^{th}term=m$

$ora+(n-1)d=m.....\to \left(2\right)$

$from\left(1\right)and\left(2\right),weget(m-n)d=(n-m)$

$\therefore d=-1$

now

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

$\therefore a=n+m-1$

hence

$p^{th}term=a+(p-1)d$

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

$=n+m-p$

and

$(m+n{)}^{th}term=a+(m+n-1)d$

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

$=0$