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
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.
Memorial Middle School