Download Fibonacci Extended

InterMath | Workshop Support | Write Up Template
March 30, 2004
Title
Fibonacci Extended
Problem Statement
Choose two whole numbers (number one and number two). Add them together and form a
Fibonacci-like sequence (add number one and number two together to get number three,
then add number two and number three together to get number four, etc.). End with a total
of ten numbers. Repeat the process by starting with two different numbers.
What is the relationship between the seventh term and the sum of the terms (for each
sequence)? What is the relationship between the seventh term and the tenth term (for each
sequence)? Explain.
Problem setup
I am familiar with the Fibonacci sequence and I would like to investigate more on the topic to
paint a clearer picture of the reasoning behind the pattern of numbers. The problem involves
developing a set of numbers in which number one and two can add together to get number three
and the pattern continues. After sets of numbers have been developed, it is possible to explore
the relationship between the 7th term and the sum of the numbers, as well as the 7th and 10th
terms.
Plans to Solve/Investigate the Problem
For this particular investigation, I will use Microsoft Excel to apply technology toward the
problem. Microsoft Excel will allow me to display calculations and relationships among the
numbers clearly. It is a fascinating program to use when looking at number patterns.
Investigation/Exploration of the Problem
I would like to start the investigation with a three sets of different positive, whole numbers to
determine a common relationship. All three sets with calculations are included below:
Set #1:
Sum / 7th term
2
5
7
12
19
31
50
81
131
212
10th / 7th term
Set #2:
11
8
11
19
30
49
79
128
207
335
542
4.24
550
Set #3:
11
24
30
54
84
138
222
360
582
942
1524
4.234375
1408
11
4.233333
3960
After calculating each set in Excel, I found a distinct relationship between the sum of the terms
and the 7th term. I found that in each set, the sum of the terms divided by the 7th term always
equaled 11. After reading about the Fibonacci numbers, I found that the number 11 is called the
golden string. It was neat to find that each different set of numbers shared the relationship of 11
among the sum of all numbers and the 7th term. Those calculations are highlighted in yellow on
the Excel document.
Next, I investigated the relationship between the 10th and 7th terms in the pattern. In each set, I
divided the 7th term into the 10th term and found another distinct relationship. The outcome of
each calculation was always about 4.23. I found that the 10th term was about 4 times that of the
7th term. This was an interesting relationship that worked in each set of numbers. The
calculations are highlighted in blue.
Last, I explored the golden ratio between each of the sets of numbers. The golden ratio is the
number 1.618034 or PHI (p) that develops as a pattern in the Fibonacci sequence. Basically,
each number is divided by the number before it. The golden ratio appears several times within
the sequence. When graphed, there would only be a few points that lie outside of the golden
ratio. The calculations for each set are included below:
Set #1
The Golden Ratio:
1
2.5
1.4
1.71428571
1
1.63157895
1.61290323
1.62
1
1.61832061
Set #2
1
1.375
1.72727273
1
1.63333333
1.6122449
1
1.6171875
1.61835749
1
Set #3
1
1.25
1.8
1
1.642857143
1.608695652
1
1.616666667
1.618556701
1
This was an interesting investigation that provided me with a lot of information through
exploration of the Fibonacci sequence.
Extensions of the Problem
Would your result be different if you started with negative numbers or fractions?
To extend the problem further, I completed the extension as an Excel document. All information
is included in Set 4 (negative numbers) and Set 5 (fractions). I have drawn the same conclusions
that were investigated in the original problem. The information from Excel is included below:
Set #4:
-9
-15
-24
-39
-63
-102
-165
-267
-432
-699
Set #5:
11
1/5
1/8
1/3
4/9
7/9
1 2/9
2
3 2/9
5 2/9
8 4/9
4.236364
4 2/9
-1815
22
1
1.6666667
1.6
1
1.6153846
1.6190476
1
1.6181818
1.6179775
1
1
5/8
2 3/5
1 3/8
1
1 4/7
1 5/8
1 3/5
1
1 5/8
Author & Contact
Nicole Vater
[email protected]
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