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      Question

      The sequence below has a single missing term denoted by

      “?”: 3, 5, 11, 23, 47, ? Condition I: The first difference (T₂ − T₁, T₃ − T₂, …) of the sequence forms a geometric progression. Condition II: The first three terms of the sequence are 3, 5 and 11. What is the missing term?
      A 87 Correct Answer Incorrect Answer
      B 95 Correct Answer Incorrect Answer
      C 99 Correct Answer Incorrect Answer
      D 105 Correct Answer Incorrect Answer
      E None of these Correct Answer Incorrect Answer

      Solution

      ATQ, Let the sequence be T₁, T₂, T₃, T₄, T₅, T₆: T₁ = 3 T₂ = 5 T₃ = 11 T₄ = 23 T₅ = 47 T₆ = ? First differences: d₁ = T₂ − T₁ = 5 − 3 = 2 d₂ = T₃ − T₂ = 11 − 5 = 6 d₃ = T₄ − T₃ = 23 − 11 = 12 d₄ = T₅ − T₄ = 47 − 23 = 24 Observe d₂/d₁ = 6/2 = 3 d₃/d₂ = 12/6 = 2 d₄/d₃ = 24/12 = 2 So after d₂, the ratio is 2. This suggests from d₂ onward the differences double each time, starting from 6: d₂ = 6 d₃ = 12 = 2·6 d₄ = 24 = 2·12 Thus the next difference: d₅ = 2·24 = 48 Then: T₆ = T₅ + d₅ = 47 + 48 = 95 Therefore, the missing term is 95. (Geometric progression of differences from d₂ onward: 6, 12, 24, 48,… with common ratio 2.)

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