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(a^(n+k) + b^(n+k) + a^(n-k) + b^(n-k))/(a^n + b^n) for the roots of a quadratic whose product of roots has modulus 1

The same question, asked 2 times: 11 Apr 2023*, 24 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    11 Apr 2023

    89
    If aa and bb are the roots of equation x2−7x−1=0x^{2}- 7x - 1 = 0, then the value of a21+b21+a17+b17a19+b19\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}} is equal to

  2. Question 2

    24 Jan 2025

    56
    If α\alpha and β\beta are the roots of the equation 2z2−3z−2i=02z^{2}- 3z - 2i = 0, where i=−1i =\sqrt{- 1}, then 16. Re(α19+β19+α11+β11α15+β15)⋅Im(α19+β19+α11+β11α15+β15)Re\left( \frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}} \right)\cdot Im\left( \frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}} \right) is equal to

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