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) term of the A.P. is zero

Answer 1

Let a be the first term and d be the common difference of the given A.P. Then, m times m^th term = n times n^th term

⇒ ma_m = na_n

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

⇒ m{a + (m – 1) d} – n{a + (n – 1) d} = 0

⇒ a(m – n) + {m (m – 1) – n(n – 1)} 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

⇒ a_m+n = 0

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