Assignment #12

Explorations with the Fibonacci Sequence

by

Angela Wall

Generate a Fibonacci sequence in the first column using f(0) = 1, f(1) = 1,

f(n) = f(n-1) + f(n-2).

Construct the ratio of each pair of adjacent terms in the Fibonacci sequence. What happens as n increases? What about the ratio of every second term? etc.

Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0)=1 and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.

For centuries, the sequence of Fibonacci numbers has proven to be a topic that goes much deeper than a simple list of certain numbers. The Fibonacci sequence has been connected to other realms of mathematics and several phenomena found throughout the natural world. Introducing students to the sequence and some of its elementary properties could spark some fascination with the splendor of mathematics. An exploration with the sequence can easily be done using some sort of spreadsheet software, such as Microsoft Excel.

The table below shows the Fibonacci sequence, starting with 1 as the first and second terms. This can be easily derived in Excel by adding a function to the column, where the next cell is the sum of the previous two cells. Applying the function to several cells in the column will result in the Fibonacci sequence. Observe how the numbers in the sequence increase quite rapidly.

One of the interesting properties of the Fibonacci sequence is the ratio of each pair of adjacent terms in the sequence. When dividing the n^{th} term in the sequence by the n-1 term in the sequence, the ratio converges to approximately 1.618034 as the value for n increases. This value is the Golden Ratio (or Golden Mean). When dividing the n-1 term by the n^{th} therm, the ratio converges to approximately .618034 as the value for n increases. This is the Golden Ratio minus one. Notice in the table below how the values in the column approach 1/2*(1±√5) as the value for n increases.

Let *a* and *b* be two successive terms in the sequence, where *a* is before *b*. Thus, *a+b* is the next term. When dividing the n^{th} term in the sequence by the n-1 term, essentially we are finding *b*/*a*. The table above shows us that this ratio converges to a particular number, so the ratio (*a+b*)/*b* should converge to the same number. So, *b*/*a* = (*a+b*)/*b* = *a*/*b* + 1. Let L be the limit of *b*/*a* as n increases. So, L = 1/L + 1. This results in the quadratic equation L^{2} - L - 1 = 0. Using the quadratic formula to find the solutions of this equation, L = 1/2*(1±√5), which is the Golden Ratio. (An exploration with the powers of L will be further examined below.)

Another interesting property of the Fibonacci sequence is the ratio of every second term in the sequence. When dividing the n^{th} term in the sequence by the n-2 term, the ratio converges to approximately 2.618034 as n increases, which is the Golden Ratio, plus one. When dividing the n-2 term in the sequence by the n^{th} term, the ratio converges to approximately .381967 as n increases. This is the two minus the Golden Ratio.

Other sequences with different initial values, but with the same rule as the Fibonacci sequence (adding consecutive terms to get the next term) have similar properties of that of the Fibonacci sequence. For example, the Lucas sequence is similar to the Fibonacci sequence, but has the initial values of 1 and 3. Finding the same ratios as mentioned above with the Fibonacci sequence, they all converge to the same values related to the Golden Ratio as the value for n increases.

The limit of the ratio of the n^{th} and n-1 terms and the n^{th} and n-2 terms always converge to the same values as that of the Fibonacci sequence, regardless of the two initial values. See the different tables below of different sequences that result in the same limits as n approaches infinity.

Another interesting property of the Fibonacci sequence results from certain operations with successive terms in the sequence. First, find the product of successive terms in the sequence. Using these values, find the difference between the successive terms. This is the difference of the products. Taking the square root of each difference will result in the Fibonacci sequence once again (except the first initial term). The property is true for any sequence whose terms are the sum of the two previous consecutive terms. Note that when the second initial value or both initial values are negative (making the sequence all negative values), the resulting sequence after the three operations will be all positive.

Consider the algebra to understand why this works. Let *a*, *b*, and *a+b* be three consecutive terms in the sequence. Then the products of two adjacent terms are *a*b* = *ab* and *b**(*a+b*) = *ab* + *b*^{2}. The difference between these two products is (*ab + b*^{2}) - *ab* = *b*^{2}. Taking the square root of *b ^{2}* results in

As described above, the ratio of consecutive terms in the Fibonacci sequence converged to the Golden Ratio. This was shown using *a* and *b* as consecutive terms in the sequence.

Now, consider the consecutive terms in the sequence as f_{n+1} and f_{n}. The ratio of the adjacent terms as n increases can be denoted as

Let L be the limit of f_{n+1}/f_{ n }. It follows that for large enough n, L will also be the limit of f_{n}/f_{n-1}. So the limit of f_{n-1}/f_{n} = 1/L. Substituting in values, we have L = 1/L + 1 or 1/L = L - 1. This implies that the reciprocal of the limit of consecutive terms is given by the limit of consecutive terms, minus one.

Multiplying both sides of the equation above by L results in L^{2} = L + 1. Solving this quadratic equation results in the Golden Ratio, usually denoted by Φ. So, it was found that Φ = 1/Φ + 1 and Φ^{2} = Φ + 1. The fact that Φ^{2} = Φ + 1 implies that the square of the Golden Ratio is the same as one plus the Golden Ratio.

Notice that in the explorations above, Φ + 1 was also found in the convergence of the ratio of every second term. Thus, the ratio of every second term is Φ^{2}. Similar to that above, consider the folllowing:

What about Φ^{3}? Will this be the limit of every third term in the Fibonacci sequence? Let f_{n+3} and f_{n} be every third term in the sequence.

The table below shows how the terms in the Fibonacci sequence are related to the powers of Φ. Notice that the the ratio of adjacent terms converge to Φ, the ratio of every two terms converge to Φ^{2}, the ratio of every three terms converge to Φ^{3}, the ratio of every four terms converge to Φ^{4}, etc.

Writing powers of Φ in terms of Φ also has an interesting result. Using the fact that Φ = 1 + 1/Φ, multiplying by Φ, and making appropriate algebraic substitutions results in

Φ^{ 2 } = Φ + 1

Φ^{ 3 } = 2Φ + 1

Φ^{4} = 3Φ + 2

Φ^{5} = 5Φ + 3

Φ^{6} = 8Φ + 5

Φ^{7} = 13Φ + 8

Can a pattern be observed here? Could Φ^{8} be predicted without going through the tedious substitutions and simplifications? What observations can be made about the coefficient and constant in each equation? It turns out that powers of Φ can be written as a linear function of Φ where the coefficient of Φ and the constants are adjacent terms in the Fibonacci sequence.

In other words,Φ^{n} = f_{n}Φ + f_{n-1} for n bigger than 0.

Through these various explorations, it is easy to see that the Fibonacci sequence is more than just a set of numbers. Further explorations with the sequence could allow students to discover interesting properties and outcomes that all contribute to the beauty of mathematics.