Question

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

Answer 1

According to the given condition,

⇒ mt_m = nt_n

⇒ m{a + (m – 1) d} = n {a + (n – 1) d}

⇒ ma + md(m – 1) = na + nd(n − 1) 

⇒ ma + m²d – md = na + n²d – nd 

⇒ ma + m²d – md – na – n²d + nd = 0 

⇒ (ma – na) + (m²d – n²d) – (md – nd) = 0

⇒ a(m – n) + d(m² – n²) – d(m – n) = 0 

⇒ a(m – n) + d(m + n) (m – n) – d(m – n) = 0

⇒ (m – n)[a + (m + n – 1) d] = 0

⇒ [a + (m + n – 1)d] = 0     ...[Dividing both sides by (m – n)]

⇒ t(m + n) = 0 

Hence, the (m + n)^th term of the given A.P. is zero