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  • Perplex
    IB Math AAHL
    /
    Proof and Reasoning
    /

    Problem Bank

    [Maximum mark: 13]

    Let ​Ln​​ be a sequence defined by

    ​
    ⎩⎪⎨⎪⎧​L0​=x Ln+1​=ln∣Ln​∣​
    ​

    where ​x>0​ and ​n∈Z.

    For example, ​L1​=ln∣x∣​ and ​L2​=ln∣ln∣x∣∣.

    1. Show that ​∫L0​L1​1​dx=L2​+C, where ​C∈R.

      [4]
    2. Using mathematical induction, prove that

      ​
      ∫L0​×L1​×⋯×Ln​1​dx=Ln+1​+C
      ​

      for all ​n≥1​ and any constant ​C.

      [9]

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