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QUEST 2014: Homework 6 Fibonacci Sequence and Golden Ratio

Each problem is worth 4 points.

1. Consider the following sequence: 1, 3, 9, 27, 81, 243,... .

a) What type of sequence is this?

b) What is the next term in the sequence?

c) Calculate the differences of consecutive terms below the sequence above.

d) What type of sequence is the sequence of differences you obtained in part c?

2. The ratio of the length (the longer side) to the width of a Golden Rectangle is what value? Give the name (in words), the Greek letter used to denote this value, the exact value (using a square root), and a three decimal approximation. 3. What special geometric property does a Golden Rectangle have? Draw a picture to illustrate your answer.

4. Starting with A1 = 2,A2 = 5 construct a new sequence using the Fibonacci Rule An+1 = An + An−1. Thus A3 = 5 + 2 = 7, etc..

a) Find A3,A4,...,A9.

b) Next calculate the ratios An+1 and find the value that it is ap- An proaching.

5. Explain how to draw a Golden Spiral. Illustrate. 6. Draw the rectangle with sides of integer lengths that is the best pos- sible approximation to a Golden Rectangle if a) Both sides have lengths less than 20. Calculate the ratio. Hint: Use the Fibonacci sequence.

b) Both sides have lengths less than 60. Calculate the ratio.

7. Show how to construct a Golden Rectangle starting from a square, using a straight-edge and compass.

8. Calculate the 20-th Fibonacci number with a calculator using the nearest integer formula for Fn. First use φ ≈ 1.61803399. Compare your answer using φ ≈ 1.618 instead. Explain any discrepancy you find. 9. Draw a Pentagram inside a regular Pentagon, and state two places where the Golden Ratio comes into the diagram.

10. In this exercise we find the golden angle from scratch. Start by draw- ing a circle of radius r and marking off two arcs of lengths a and b. Let a be the shorter arc, corresponding to an angle θ (in degrees), and b be the longer arc, corresponding to an angle 360 − θ. a) Express a and b in terms of θ. Recall: To get the arclength, you simply multiply the circumference of a circle by the proportion of the circle taken up by the arc.

b) Find θ so that the ratio of the longer arc to the shorter arc equals the golden ratio φ. Round your answer to two decimal places.