Question

If m times the mth term of an AP is equal to n times its nth term and m ≠ n, show that the (m + n)th term of AP is zero.

Answer 1

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

We have, mam = nan

⇒ m[a + (m – 1)d] = n[a + (n – 1 )d]

⇒ ma + m²d – md = na + n²d – nd

⇒ m²d – n²d – md + nd = na – ma

⇒ d(m² – n²) – d(m – n) = a(n – m)

⇒ d[m² – n² – (m – n)] = a(n – m)

⇒ d[(m + n) (m – n) – (m – n)] = a(m – n)

⇒ d[(m – n) (m + n – 1)] = – a(m – n)

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

⇒ d(m + n – 1) = – a

⇒ d = \(-\frac{a}{m+n+1}\)

Now, a_(m+n) = a + (m + n – 1)d

= a + {(m + n – 1) × \(\frac{-a}{(m+n-1)}\)}

= a – a = 0

Proved