Question

if m times the mth term is equal to n times the nth term of an A.P. prove that (m+n)th term of A.P is zero.

Answer 1

Let the first term of AP = a
common difference = d
We have to show that (m+n)th term is zero or a + (m+n-1)d = 0

mth term = a + (m-1)d
nth term = a + (n-1) d

Given that m{a +(m-1)d} = n{a + (n -1)d}
⇒ am + m²d -md = an + n²d - nd
⇒ am - an + m²d - n²d -md + nd = 0
⇒ a(m-n) + (m²-n²)d - (m-n)d = 0
⇒ a(m-n) + {(m-n)(m+n)}d -(m-n)d = 0
⇒ a(m-n) + {(m-n)(m+n) - (m-n)} d = 0
⇒ a(m-n)  + (m-n)(m+n -1) d  = 0
⇒ (m-n){a + (m+n-1)d} = 0 
⇒ a + (m+n -1)d = 0/(m-n)
⇒ a + (m+n -1)d = 0

Proved!

Answer 2

hence proved that (m+n)=0