Question

Let t₁ , t₂ , … be real numbers such that t₁ + t₂ +……+ t_n = 2n² + 9n + 13, for every positive integer n ≥ 2. If t_k = 103, then k equals

Answer 1

Calculation:

Given that, t₁ + t₂ +……+ t_n = 2n² + 9n + 13       ……. (1)

⇒ t₁ + t₂ +……+ t_n-1 = 2(n - 1)² + 9(n – 1) + 13     ………(2)

Equation (1) – Equation (2),

We get t_n = 2n² + 9n + 13 - 2(n - 1)² + 9(n – 1) + 13 = 4n + 7

According to the Question, t_k = 103

⇒ t_k = 4k + 7 = 103

⇒ 4k = 96

⇒ k = 24