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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] 11