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      Question

      A sequence {Sₙ} is defined by: Condition I:

      S₁ = 3 and for all n ≥ 2, Sₙ = 2·Sₙ₋₁ + (−1)ⁿ. Condition II: The second term S₂ is 7. The first six terms are: 3, 7, 13, 27, ?, 107 What is the 5th term?
      A 47 Correct Answer Incorrect Answer
      B 67 Correct Answer Incorrect Answer
      C 53 Correct Answer Incorrect Answer
      D 27 Correct Answer Incorrect Answer
      E None of these Correct Answer Incorrect Answer

      Solution

      From Condition I and S₁ = 3: For n = 2: S₂ = 2·S₁ + (−1)² = 2·3 + 1 = 7 (matches Condition II) For n = 3: S₃ = 2·S₂ + (−1)³ = 2·7 − 1 = 14 − 1 = 13 For n = 4: S₄ = 2·S₃ + (−1)⁴ = 2·13 + 1 = 26 + 1 = 27 For n = 5: S₅ = 2·S₄ + (−1)⁵ = 2·27 − 1 = 54 − 1 = 53 For n = 6: S₆ = 2·S₅ + (−1)⁶ = 2·53 + 1 = 106 + 1 = 107 (matches given) So the 5th term is 53.

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