If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.

Question

Philoid · Accepted Answer

Given: 10 × a₁₀ = 15 × a₁₅

To show : a₂₅ = 0

Consider 10 × a₁₀ = 15 × a₁₅

⇒ 10 [a + (10 - 1)d] = 15 [a + (15 - 1)d]

⇒ 10a + 90d = 15a + 210d

⇒ - 5 a = 120 d

⇒ a = - 24d ………………(1)

Now, a₂₅ = a + (25 - 1)d

⇒ a₂₅ = a + 24d

⇒ a₂₅ = - 24d + 24d (from equation 1)

⇒ a₂₅ = 0

Hence, proved.