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      Question

      A sequence {Sₙ} is defined for n ≥ 1 as follows:

      I) For every n ≥ 3: Sₙ = Sₙ₋₂ + 6 It is also known that - S₃ + S₄ = 40 and S₅ − S₂ = 10. What is the value of S₇?
      A 39 Correct Answer Incorrect Answer
      B 53 Correct Answer Incorrect Answer
      C 31 Correct Answer Incorrect Answer
      D 46 Correct Answer Incorrect Answer
      E None of these Correct Answer Incorrect Answer

      Solution

      Let S₁ = u, S₂ = v. From the recurrence: S₃ = S₁ + 6 = u + 6 S₄ = S₂ + 6 = v + 6 S₅ = S₃ + 6 = u + 12 S₇ = S₅ + 6 = u + 18 Now use the given conditions. S₃ + S₄ = 40 (u + 6) + (v + 6) = 40 u + v + 12 = 40 u + v = 28 …(i) S₅ − S₂ = 10 (u + 12) − v = 10 u − v + 12 = 10 u − v = −2 …(ii) Add (i) and (ii): (u + v) + (u − v) = 28 + (−2) 2u = 26 u = 13 Then from (i): 13 + v = 28 → v = 15 Now: S₇ = u + 18 = 13 + 18 = 31

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