Mathematics Education

EMAT 6680, Professor Wilson

Exploration 12, the Fibonacci sequence by Ursula Kirk

Generate a Fibonacci sequence in the first column using

f(0) = 1 and f(1) = 1

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

a.      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?

We can obtain the Fibonacci sequence by using the recursive formula below when

We continue constructing the sequence the same way; by the time we get to  we have the sequence as shown below. In this sequence, our numbers increase very rapidly as every term is the sum of the two terms that came before.

 f(0) 0 f(1) 1 f(2) 1 f(3) 2 f(4) 3 f(5) 5 f(6) 8 f(7) 13 f(8) 21 f(9) 34 f(10) 55 f(11) 89 f(12) 144 f(13) 233 f(14) 377 f(15) 610 f(16) 987 f(17) 1597 f(18) 2584 f(19) 4181 f(20) 6765

Next, we explore the sequence that we obtain from the ratio of the adjacent terms of the Fibonacci sequence.

 f(0) 0 f(1) 1 f(2) 1 1 f(3) 2 2 f(4) 3 1.5 f(5) 5 1.666667 f(6) 8 1.6 f(7) 13 1.625 f(8) 21 1.615385 f(9) 34 1.619048 f(10) 55 1.617647 f(11) 89 1.618182 f(12) 144 1.617978 f(13) 233 1.618056 f(14) 377 1.618026 f(15) 610 1.618037 f(16) 987 1.618033 f(17) 1597 1.618034 f(18) 2584 1.618034 f(19) 4181 1.618034 f(20) 6765 1.618034

In this sequence, we can see that as n gets larger, the ratio of each pair of adjacent terms of the Fibonacci sequence converges to 1.61803399 which is the Golden Mean.

Next, we explore the sequence that we obtain from the ratio of every second term of the Fibonacci sequence.

 f(0) 0 f(1) 1 f(2) 1 1 f(3) 2 2 2 f(4) 3 1.5 3 f(5) 5 1.666667 2.5 f(6) 8 1.6 2.666667 f(7) 13 1.625 2.6 f(8) 21 1.615385 2.625 f(9) 34 1.619048 2.615385 f(10) 55 1.617647 2.619048 f(11) 89 1.618182 2.617647 f(12) 144 1.617978 2.618182 f(13) 233 1.618056 2.617978 f(14) 377 1.618026 2.618056 f(15) 610 1.618037 2.618026 f(16) 987 1.618033 2.618037 f(17) 1597 1.618034 2.618033 f(18) 2584 1.618034 2.618034 f(19) 4181 1.618034 2.618034 f(20) 6765 1.618034 2.618034

In this sequence, we can see that as n gets larger, the ratio of every second terms of the Fibonacci sequence converges to 2.61803399 which is the Golden Mean +1.

b.      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.

The arbitrary integers I choose are  and  . Then I obtain the following sequesnce:

 f(0) 4 f(1) 8 f(2) 12 f(3) 20 f(4) 32 f(5) 52 f(6) 84 f(7) 136 f(8) 220 f(9) 356 f(10) 576 f(11) 932 f(12) 1508 f(13) 2440 f(14) 3948 f(15) 6388 f(16) 10336 f(17) 16724 f(18) 27060 f(19) 43784 f(20) 70844

Next, I will create two more sequences. One is the ratio of each pair of adjacent terms in the Fibonacci sequence. The second sequence is the ratio of every second term.

 f(0) 4 1 f(1) 8 f(2) 12 1.5 f(3) 20 1.666667 2.5 f(4) 32 1.6 2.666667 f(5) 52 1.625 2.6 f(6) 84 1.615385 2.625 f(7) 136 1.619048 2.615385 f(8) 220 1.617647 2.619048 f(9) 356 1.618182 2.617647 f(10) 576 1.617978 2.618182 f(11) 932 1.618056 2.617978 f(12) 1508 1.618026 2.618056 f(13) 2440 1.618037 2.618026 f(14) 3948 1.618033 2.618037 f(15) 6388 1.618034 2.618033 f(16) 10336 1.618034 2.618034 f(17) 16724 1.618034 2.618034 f(18) 27060 1.618034 2.618034 f(19) 43784 1.618034 2.618034 f(20) 70844 1.618034 2.618034

It is very interesting to observe that even though the values of my own “Fibonacci” sequence are different  to the real Fibonacci sequence, the third column still approaches the Golden Mean, and the fourth column also

approaches the Golden Mean +1.

The Lucas sequence has the values of , again the third column approaches the Golden Mean and the fourth column approaches the Golden Mean + 1

 f(0) 1 f(1) 3 f(2) 4 1.333333 f(3) 7 1.75 2.333333 f(4) 11 1.571429 2.75 f(5) 18 1.636364 2.571429 f(6) 29 1.611111 2.636364 f(7) 47 1.62069 2.611111 f(8) 76 1.617021 2.62069 f(9) 123 1.618421 2.617021 f(10) 199 1.617886 2.618421 f(11) 322 1.61809 2.617886 f(12) 521 1.618012 2.61809 f(13) 843 1.618042 2.618012 f(14) 1364 1.618031 2.618042 f(15) 2207 1.618035 2.618031 f(16) 3571 1.618034 2.618035 f(17) 5778 1.618034 2.618034 f(18) 9349 1.618034 2.618034 f(19) 15127 1.618034 2.618034 f(20) 24476 1.618034 2.618034