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Fibonacci Extended

Problem Statement
Choose two integers. Add them together to create a third integer. Add the second and third integer of your list to create a fourth. Continue adding the last two integers to generate a Fibonacci-like sequence, ending with a total of ten integers. Repeat the process with two different starting integers.

What is the relationship between the seventh term and the sum of the sequence? What is the relationship between the seventh and tenth term of your sequence? Explain.

Problem setup

I am assigned to choose two sets of integers and perform a series of addition operations.  The specified  sequence of operations should create a Fibonacci-like sequence, to a total of ten integers in the sequence.  I am then asked to evaluate the relationship between the seventh tern and the sum of the sequence and the seventh term and the tenth term of the sequence.

Plans to Solve/Investigate the Problem

My initial investigation involved a "quick" study of the Fibonacci sequence.

Fibonacci Sequence:  The sequence 1, 1, 2, 3, 5, 8, 13, 21, ..., where each term is the sum of the two preceeding terms.
 If we denote the nth term by fn, then by the definition fn+1 = fn + fn-1. The 0th term is accepted as 1. By our rule one can produce all the terms in the sequence.

It seems like I could understand what this is depicting once I begin working with a sequence of numbers.  Then, I "Asked Jeeves"...

 A sequence of numbers in which each number equals the sum of the two preceding numbers

Further investigation of this problem lies in the actual sequencing of numbers as specified in the problem.  I will use Excel to accomplish this part of the problem.  A spreadsheet will likely make the part of the problem where relationships must be evaluated must easier.

Investigation/Exploration of the Problem

Here is the spreadsheet of the two sequences:

 Case 1 Case 2 Integer 1: 2 1 Integer 2: 4 3 Integer 3: 6 4 Integer 4: 10 7 Integer 5: 16 11 Integer 6: 26 18 Integer 7: 42 29 Integer 8: 68 47 Integer 9 110 76 Integer 10: 178 123 Sum of the Sequence 462 319

This was the easy part.  The remainder of the investigation involves discovering the relationships between the seventh term and the sum of the sequence, and the seventh term and the tenth term of the sequence.

I first checked to see if the sum of the sequence was divisible by the 7th term.

 Case 1: Case 2: 462 = 11 319 = 11 42 29

Why might this be?

The relationship between the seventh and tenth integers in the sequence appears to be a much more challenging one to assess.

I decided to check to see if it matters, for the purpose of this problem, if the integers selected as the 1st and 2nd integers of the sequence are consecutive.

 Case 3 Case 4 Integer 1: 2 6 Integer 2: 3 7 Integer 3: 5 13 Integer 4: 8 20 Integer 5: 13 33 Integer 6: 21 53 Integer 7: 34 86 Integer 8: 55 139 Integer 9 89 225 Integer 10: 144 364 Sum of the Sequence 374 946

I discovered that the relationship between the 7th term and the sum of the sum of the sequence, even when the selected integers are sequential, is still evenly divisible by 11.

 Case 3: Case 4: 374 = 11 946 = 11 34 86

I played around with some numbers to see if the 7th digit was some multiple of the 10th number in the series.  I found that the 7th digit multiplied by 4 PLUS the 4th digit equals the 10th digit!!

CAUTION:  THIS IS MERELY A CHANCE DISCOVERY!!!

Let's check it out for all four of the cases:

 Case 1 Case 2 Case 3 Case 4 Integer #7 42 29 34 86 Integer #7 times 4 168 116 136 344 Integer #7 times 4 + the 4th Digit 178 123 144 364 Equals 10th digit? yes yes yes yes

Algebraically, this looks like 4(n7) + n4 = n10?

Whew!!!

Author & Contact
Angela M. Gilliam
agilliam@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials