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Function f(n)

Source: AHSME 1964 Problem 31

January 13, 2014
functionalgebrapolynomialAMC

Problem Statement

Let f(n)=5+3510(1+52)n+5−3510(1−52)n.f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n. Then f(n+1)−f(n−1)f(n+1)-f(n-1), expressed in terms of f(n)f(n), equals:
<spanclass=′latex−bold′>(A)</span> 12f(n)<spanclass=′latex−bold′>(B)</span> f(n)<spanclass=′latex−bold′>(C)</span> 2f(n)+1<spanclass=′latex−bold′>(D)</span> f2(n)<spanclass=′latex−bold′>(E)</span> 12(f2(n)−1)<span class='latex-bold'>(A)</span>\ \dfrac{1}{2}f(n) \qquad <span class='latex-bold'>(B)</span>\ f(n)\qquad <span class='latex-bold'>(C)</span>\ 2f(n)+1 \qquad <span class='latex-bold'>(D)</span>\ f^2(n) \qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{2}(f^2(n)-1)
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