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? asked in Science & MathematicsMathematics · 5 years ago

Prove mathematically that φ (golden ratio ~ 1.618) sum of decrementing power that is φ ^ n = φ ^ (n - 1) + φ ^ (n - 2)?

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  • Nick
    Lv 6
    5 years ago

    φ ^ n = φ ^ (n - 1) + φ ^ (n - 2)

    φ^2 = φ + 1

    φ^2 - φ - 1 = 0

    φ = (1 +- sqrt(5))/2

    The positive solution is the golden ratio (1+sqrt(5))/2 ~ 1.618.

  • Barry G
    Lv 7
    5 years ago

    By definition

    φ = a/b = (a+b)/a

    so

    φ = 1+1/φ

    φ^2 = φ+1.

    Multiply by φ^(n-2) :

    φ^n = φ^(n-1)+φ^(n-2).

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