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      Question

      Each question below is based on a number series.

      Identify the pattern and answer accordingly. A sequence {Aₙ} is defined as follows: A₁ = p (unknown) For n ≥ 2: Aₙ = 2Aₙ₋₁ + c, where c is a constant (same for all n). It is given that: A₃ = 50 and A₄ = 106. What is the value of A₅?
      A 211 Correct Answer Incorrect Answer
      B 227 Correct Answer Incorrect Answer
      C 218 Correct Answer Incorrect Answer
      D 233 Correct Answer Incorrect Answer
      E None of these Correct Answer Incorrect Answer

      Solution

      Given recurrence: Aₙ = 2Aₙ₋₁ + c Start from A₁ = p: A₂ = 2p + c A₃ = 2A₂ + c = 2(2p + c) + c = 4p + 3c A₄ = 2A₃ + c = 2(4p + 3c) + c = 8p + 7c Given: A₃ = 50 → 4p + 3c = 50 …(i) A₄ = 106 → 8p + 7c = 106 …(ii) Solve (i) & (ii): Multiply (i) by 2: 8p + 6c = 100 …(iii) Subtract (iii) from (ii): (8p + 7c) − (8p + 6c) = 106 − 100 c = 6 Substitute in (i): 4p + 3×6 = 50 → 4p + 18 = 50 → 4p = 32 → p = 8 Now: A₄ is already 106 (given). A₅ = 2A₄ + c = 2×106 + 6 = 212 + 6 = 218

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