The numbers that appear in the explicit formula for the

Chapter 5, Problem 24E

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The numbers that appear in the explicit formula for the Fibonacci sequence are related to a quantity called the golden ratio in Greek mathematics. Consider a rectangle of length ? units and height 1, where ? > 1. Divide the rectangle into a rectangle and a square as shown in the preceding diagram. The square is 1 unit on each side, and the rectangle has sides of lengths 1 and ? ? 1.The ancient Greeks considered the outer rectangle to be perfectly proportioned (saying that the lengths of its sides were in a golden ratio to each other) if the ratio of the length to the width of the outer rectangle equaled the ratio of the length to the width of the inner rectangle. That is, .a. Show that ? satisfies the following quadratic equation: b. Find the two solutions of and call them?1 and ?2.c. Express the explicit formula for the Fibonacci sequence in terms of ?1 and ?2.