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Let alpha and beta be the roots of the quadratic equation x² - 4x + 1 = 0. Define the sequence aₙ = alphaⁿ + betaⁿ. Find the value of (a₂₀₁₇ + a₂₀₁₅) / a₂₀₁₆.

1. 4
2. 2015
3. 4²⁰¹⁶
4. 1/4

Correct answer: 4

Solution

Because alpha and beta are roots of x² - 4x + 1 = 0, they satisfy x² = 4x - 1. Multiplying alpha^(n-1) + beta^(n-1) through gives the recurrence a_(n+1) = 4*aₙ - a_(n-1), which rearranges to a_(n+1) + a_(n-1) = 4*aₙ. Setting n = 2016 yields (a₂₀₁₇ + a₂₀₁₅)/a₂₀₁₆ = 4.

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