The 4th term of an AP is zero. Prove that its 25th term is triple its 11th term.

Question

Philoid · Accepted Answer

Let a be the first term and d be the common difference.

Given: a₄ = 0

To prove: a₂₅ = 3 × a₁₁

Now, Consider a₄ = 0

⇒ a + (4 - 1)d = 0

⇒ a + 3d = 0

⇒ a = - 3d ………. (1)

Consider a₂₅ = a + (25 - 1)d

⇒ a₂₅ = - 3d + 24d (from equation (1))

⇒ a₂₅ = 21d ………. (2)

Now, consider a₁₁ = a + (11 - 1)d

⇒ a₁₁ = - 3d + 10d (from equation (1))

⇒ a₁₁ = 7d ……….(3)

From equation (2) and (3), we get,

a₂₅ = 3 × a₁₁

Hence, proved.