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 Fibonaccilike 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 Fibonaccilike 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 n^{th}
term by f_{n}, then by the definition f_{n+1} =
f_{n} + f_{n1}. The 0^{th} 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"...
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(n_{7})
+ n_{4} = n_{10}?
Whew!!!
Author & Contact
Angela M. Gilliam
agilliam@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
Click here to
view the original problem at InterMath
