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.

 

Investigation/Exploration of the Problem

 

Let's choose the integers 1 and 3. If we add these together, we get 4. So our sequence would be the following: 1, 3, 4, 7, 10, 17, ......

Now let's use 2 and 5. If we add these together, we get 7. So our sequence would be: 2, 5, 7, 12, .....

Click here to see the spreadsheet of these sequences.

 

From the spreadsheet we see that the sum of the sequence is equal to the seventh term multiplied by 11. The same is true when we multiply the seventh term in Fibonacci's Sequence i.e. 13*11 = 143.

 

Now the tough part....What is the relationship between the seventh and tenth term of your sequence? This is not obvious. After collaborating with two classmates, we found that if you multiply the seventh term by four and then subtract the answer from the tenth term, you are left with the fourth term. For example, in the first sequence the seventh term is 102, the tenth term is 432 and the fourth term is 24. Using our "formula" we find the following:

 

102*4 = 408

432-408 = 24

 

First sequence
6
9
15
24
39
63
102
165
267
432

 

Click here for the spreadsheet

 

Extensions of the Problem

Would your result be different if you started with negative numbers or fractions?

To find out we simply plug in the the negative number and the fractions in the spreadsheet and we see that the relationship holds true. Click here for the spreadsheet.

 

Author & Contact
Laura Thomas

Memorial Middle School

Conyers, GA
lthomas@rockdale.k12.ga.us