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      Question

      A sequence {Tₙ} is defined as follows:

      Condition I: T₁ = 2 and for all n ≥ 2, Tₙ = 2·Tₙ₋₁ + (2n − 1). Condition II: T₄ = 45 (this is given as a consistency check). The first six terms are of the form: 2, 7, ?, 45, ?, 209 What is the 5th term of the sequence?
      A 87 Correct Answer Incorrect Answer
      B 117 Correct Answer Incorrect Answer
      C 99 Correct Answer Incorrect Answer
      D 77 Correct Answer Incorrect Answer
      E None of these Correct Answer Incorrect Answer

      Solution

      ATQ, From Condition I: T₁ = 2 For n = 2: T₂ = 2·T₁ + (2·2 − 1) = 2·2 + 3 = 7 For n = 3: T₃ = 2·T₂ + (2·3 − 1) = 2·7 + 5 = 14 + 5 = 19 For n = 4: T₄ = 2·T₃ + (2·4 − 1) = 2·19 + 7 = 38 + 7 = 45 (matches Condition II) For n = 5: T₅ = 2·T₄ + (2·5 − 1) = 2·45 + 9 = 90 + 9 = 99 For n = 6: T₆ = 2·T₅ + (2·6 − 1) = 2·99 + 11 = 198 + 11 = 209 (matches given) So the 5th term is 99.

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