Question

If the 9th term of an AP is zero, then prove that its 29th term is twice its 19th term.

Answer 1

Let the first term, common difference and number of terms of an AP are a, d and n respectively.

Given : 9^th term is zero i.e. a₉ = 0

To prove : a₂₉ = 2a₁₉

Proof :

As a₉ = 0

a + 8d = 0 [ eqn i]

Using the nth term formula i.e. a_n = a + (n - 1)d

Taking LHS

a₂₉ = a + 28d

= (2 - 1)a + (36 - 8)d

= 2a - a + 36d - 8d

= 2a + 36d - (a + 8d)

= 2(a + 18d) - 0 [ using i]

= 2a₉ [ as a₉ = a + 8d]

= RHS