give an easy solution for this question:

if m times the m^th term of an A.P is equal to n times its n^thterm, show that {mthe A+n}^th term of A.P.

is zero

Let a be the first term and d be the common difference.Then
m th term of A.P is a+(m-1)d

n th term of A.P is a+(n-1)d.
Given m(a+(m-1)d) = n(a+(n-1)d)
ma+m(m-1)d = na+n(n-1)d
(m-n)a + (m^2-n^2-m+n)d=0
(m-n)a+( (m-n)(m+n)-(m-n) )d=0
(m-n)(a+(m+n-1))d=0
a+(m+n-1)d=0.
This is the (m+n)th term of A.P.