Question

If m times m^th term of an A.P. is equal to n times its n^th term, then show that (m + n)^th term of the A.P. is zero.

Answer 1

Given:-

t_m = [a+(m−1)d]

t_n = [a+(n−1)d]

m ( t_m) = n (t_n)

m[a+(m−1)d] = n[a+(n−1)d]

⇒ m[a+md−d] = n[a+nd−d]

⇒am+m²d−md = an+n²d−nd

⇒ am+m²d−md−an−n²d+nd = 0

⇒ am−an+ m²d− n²d−md+nd = 0

⇒ a(m−n)+d(m^₂− n^₂) −d(m−n) = 0

⇒ (m-n)[a+d(m+n)-d] = 0 … [Divide by (m−n)]

⇒ [a+d(m+n)-d] = 0

⇒ a+(m+n−1)d = 0

⇒ t_m+n = 0