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    The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(n+1) + F_(n-1) for each n > or equal to 2. Prove that L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

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    Theory of Numbers (XIV)
    Principle of Mathematical Induction
    Fibonacci Number
    Lucas number

    The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(n+1) + F_(n-1) for each n > or equal to 2.
    Prove that
    L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

    See the attached file.

    © BrainMass Inc. brainmass.com March 4, 2021, 8:09 pm ad1c9bdddf
    https://brainmass.com/math/number-theory/148314

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    Theory of Numbers (XIV)
    Principle of Mathematical Induction
    Fibonacci Number
    ...

    Solution Summary

    This solution is comprised of a detailed explanation of the Lucas numbers L_n . It contains step-by-step explanation of the Lucas numbers L_n defined by the equations L_1 = 1 and
    L_n = F_(n+1) + F_(n-1) for each n > 2 and prove of the equation

    L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

    Solution contains detailed step-by-step explanation. Note is also given at end.

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