# Assignment #12

## Golden Fibonacci Numbers

### Introduction

In this write-up, I discuss the exporation of the properties of certain recursive sequences including the Fibonacci sequence using a spreadsheet as a tool.

### The Fibonacci Sequence

Consider a student learning about arithmetic and geometric progressions. The student will develop an understanding of the relationship between the nth term of the sequence and the common difference or ratio. Sequences are recursive, which means that subsequent terms can be related to terms found earlier in the sequence. For sequences defined as arithmetic or geometric, a student can develop a formula to find the nth term of a sequence knowing only the first term and the common difference or ratio between terms. This formula is known as an explicit formula for the nth term.

But what about sequences that are neither arithmetic or geometric? The most famous infinite sequence that is neither arithmetic or geometric is the Fibonacci sequence. Because it is recursive, consecutive terms can be related to previous terms in the sequence. The recursive formula for the nth term of the Fibonacci sequence is:

## F(n) = F(n-1) + F(n-2), where F(0) = 1 and F(1) = 1

Faced with this calculation, students can see that the value of F(n) becomes hard to compute rapidly as n increases from zero. Using a spreadsheet, a student can generate large terms from the Fibonacci sequence with relative ease.

Consider the question, what is the 50th term of the Fibonacci sequence? The manual approach will be very tedious and time consuming. To manipulate a spreadsheet to find the answer, students will enter the values for F(0) and F(1) into the first two rows of the first column. Next they will create a formula in the third row of the column that adds the values in the previous two rows. This formula can be quickly copied to any number of n subsequent rows in the first column, and the student now has values for the first n terms of the sequence.

Using Microsoft Excel, the following table provides the values for the 44th - 54th terms of the Fibonacci sequence.

 Term Fibonacci numbers 44 1134903170 45 1836311903 46 2971215073 47 4807526976 48 7778742049 49 12586269025 50 20365011074 51 32951280099 52 53316291173 53 86267571272 54 139583862445

Note that the 54th term is the first term in the sequence to have 12 digits. Further explorations of the sequence with the spreadsheet could lead to answers of related questions such as:

What is the first term in the sequence to have 20 digits?

Is there a relationship between the number of consecutive terms that have equal numbers of digits?

An inquisitive mind might wonder, what happens to the relationship between the consecutive terms of the Fibonacci sequence as the number of terms approaches infinity? It is easy to see the relationship between terms using Excel by simply dividing the n term by the n-1 term. Students can see that the ratio between terms begins to approach a certain value early in the sequence. This value is the Golden Ratio, or Phi, which is equal to approximately 1.618. The table below shows the first 16 terms of the sequence and the ratio between the terms approaching 1.618.

 Term Fibonacci numbers Ratio of terms 0 1 1 1 1 2 2 2 3 3 1.5 4 5 1.66666666666667 5 8 1.6 6 13 1.625 7 21 1.61538461538462 8 34 1.61904761904762 9 55 1.61764705882353 10 89 1.61818181818182 11 144 1.61797752808989 12 233 1.61805555555556 13 377 1.61802575107296 14 610 1.61803713527851 15 987 1.61803278688525

Further exploration leads to questioning, what about the ratio between every other term? Using a spreadsheet it can be seen that this ratio approaches approximately 2.618, which is the square of the Golden Ratio and the Golden Ratio plus 1. To view a spreadsheet containing the first 250 terms of the Fibonacci sequence with the ratio between each term and every other term calculated, click here.

Within the context of sequence terminology, the student might notice that as n grows larger, the sequence is approaching a geometric sequence, because the ratio between terms is approaching a constant value. This leads to the question, is there an explicit formula for the nth term of the Fibonacci sequence?

### Related Sequences

Because the term arithmetic sequence denotes a different mathematical concept, for the purpose of this discussion we will consider the Fibonacci sequence to be a recursive adding sequence, denoting that the sequence is recursive and that the consecutive terms are sums of the previous terms. Our exploration of the Fibonacci sequence demonstrates that the ratio between terms approaches the Golden Ratio as n approaches infinity. Using a spreadsheet, students can further explore recursive adding sequences other than the Fibonacci. The challenge is to answer the question - given the formula for a recursive sequence:

## F(n) = F(n-1) + F(n-2)

What happens when F(0) and F(1) are values other than 1, as in the Fibonacci sequence?

Using a spreadsheet, students can see that the ratio between terms of any recursive sequence defined by the above equation is the Golden Ratio. First we will consider the sequence with F(0) = 1 and F(1) = 3, known as the Lucas Sequence. The table below shows the first 16 terms of this sequence and the ratio between the terms.

 Term Lucas Sequence Ratio of terms 0 1 1 3 3 2 4 1.33333333333333 3 7 1.75 4 11 1.57142857142857 5 18 1.63636363636364 6 29 1.61111111111111 7 47 1.62068965517241 8 76 1.61702127659574 9 123 1.61842105263158 10 199 1.61788617886179 11 322 1.61809045226131 12 521 1.61801242236025 13 843 1.61804222648752 14 1364 1.61803084223013 15 2207 1.61803519061584

We can see that the ratio between terms again approaches the Golden ratio. What about other values of F(0) and F(1)? Further exploration leads to the conjecture that any sequence that follows the formula

## F(n) = F(n-1) + F(n-2)

will approach a ratio between terms that is the Golden Ratio. To see a spreadsheet showing the first 100 values of the Lucas sequence, a recursive adding sequence with F(0) = 1 and F(1) = 10, and a recursive adding sequence with F(0) = 10 and F(1) = 100, click here.

The students can then explore the answers to a number of questions for different starting values of the sequence:

• After how many terms in the sequence will the ratio between the terms reach 1.618, and after that term the first three significant digits of the ratio do not change as n grows larger? For example, this occurs at the 12th term of the Fibonacci sequence.
• How does the this value change as the values of F(0) and F(1) change?
• Can you find values of F(0) and F(1) that maximize the number of terms before the ratio stabilizes at 1.618?

These and other questions generated from exploration can be answered using a spreadsheet.