Question

The nth term of an A.P. is n and its nth term is m. Prove that its pth term is $m+n-p$. Also show that its $(m+n)$th term is zero.

Answer 1

$T_m=a+(m-1)d=n,T_n=a(n-1)d=m$
Solving, we get
$d=-1$ and $a=m+n-1$.
$\therefore T_p=a+(p-1)d$
$=m+n-1+(p-1)(-1)=m+n-p$.
$T_{m+n}=a+(m+n-1)d$
$=(m+n-1)+(m+n-1)(-1)=0$.