In an AP if m times the mth term is equal to n times the nth term,then find (m+n)th term

 solution

Let a and d is first term and common difference of AP.

According to Question,

m times the mth term is equal to n times the nth term

man = nan

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

ma + m^2d - d = na+n^2d -nd

ma -na + m^2d - n^2d  = md -nd

a(m - n) + d(m^2 - n^2)=d(m - n)

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

a(m - n) =d{(m - n) - (m+n)(m-n)}

a(m - n) =d(m - n){1 - (m+n)}

a=d{1 - (m+n)}

now, (m+n)th term is 

a(m+n)  = a + (m+n-1)d

a(m+n)  = d{1 - (m+n)}+ (m+n-1)d

a(m+n)  = d - md - nd+ md + nd - d

a(m+n)  =0