Name: ___________________

Math 122 Lab 3: Golden Calculations

Introduction

In mathematical terms, infinite series is all about finding limits of infinite calculations. In this lab, you will look at other types of infinite calculations. Along the way, you will be introduced to two of the most famous objects in mathematics, the golden mean and the Fibonacci sequence.

What to Turn In

Turn in this worksheet with everything filled in. This will be graded for accuracy and understanding, so write good explanations when you are asked to explain something. Due Friday at the beginning of class.

The Golden Mean φ

Use the quadratic formula to solve the equation . The positive solution is called φ (the golden mean). Write out the exact value of φ and also its decimal approximation.

The Fibonacci Sequence

The Fibonacci sequence starts with F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5, F₆ = 8, and so on with the next term being the sum of the previous two. This is written F_n+2 = F_n+1 + F_n . Write out the terms F₇ → F₁₀.

Next, compute ratios of consecutive terms written above. That is, compute decimal approximations of , and . Use these calculations to conjecture the limit .

Another famous sequence is the Lucas sequence L₁ = 2, L₂ = 1, L₃ = 3, L₄ = 4, L₅ = 7, L₆ = 11, and so on with the next term being the sum of the previous two. Write out the terms L₇ → L₁₀ and conjecture the limit .

We can prove that any additive sequence has the same limiting ratio. Assume that S₁, S₂, S₃, S₄ and so on is a sequence with S_n+2 = S_n+1 + S_n . Divide each term in this equation by S_n+1 and then take the limit as n goes to ∞. If we name the limit , explain why . Then multiply the equation by R and explain why R = φ.

A special version of this type of sequence starts with S₁ = 1 and S₂ = φ. Write out S₃ and S₄ in exact form and as decimals.

Define the (geometric) sequence with G₁ = 1, G₂ = φ, G₃ = φ², G₄ = φ³ and so on. Write out the decimal approximations of G₃ and G₄ and compare to the S sequence.

Prove that S₃ = G₃. (Hint: Use the exact forms for both and use the quadratic equation that defines φ.)

The Fibonacci sequence and golden mean in nature

If you look at a pineapple’s bumps, you can see spirals forming in two directions shown below.

For the pineapple in class, count the number of spirals in each direction. Is this a Fibonacci number?

The spirals on a sunflower are also Fibonacci numbers, as shown. Surprising, isn’t it?

Some other facts: many flowers have 3 or 5 petals, but bloodroots have 8 petals, black-eyed Susans have 13 petals, ordinary daisies have 34 petals, but exotic varieties have 21, 55 and even 89 petals.

The golden mean is commonly used in architecture and art. The Parthenon has a rectangular outline where the ratio of the sides is almost exactly φ. The Greeks called this a “golden rectangle.” DaVinci’s paintings are now famous for having precise dimensions based on φ, which was called the “Divine Proportion” in Renaissance days.

Continued Fractions

Historically, a number of different calculation techniques competed with the techniques we now know as algebra and calculus. Continued fractions were one popular technique. Look at the sequence below:

1, , , , and so on.

Simplify each term to a simple fraction. (The second one is , the third one is and so on.)

What do you notice about the values of the numerators and denominators? Based on this, what is the limit of this sequence?

Another way of finding the limit of this continued fraction is to notice the repetitive nature of the fraction. If we name L = notice that the stuff underneath the first fraction bar looks exactly like L. In other words, we have L = . Solve this equation.

An exact formula

For the last part of the assignment, we return to the Fibonacci sequence F_n. It can be shown that where γ is the negative solution found on page 1. Verify this formula on your calculator for n = 6 and n = 7. Then use the formula to compute F₂₀ and F₅₀.

Is it surprising that a formula like this always produces an integer?

The last problem of this lab is to show that this formula satisfies the Fibonacci additive relationship. That is, show that . (Hints: Start on the left side of the equation and factor. Then use the quadratic equation that both φ and γ satisfy and you should get the right side as desired.)