Suppose you place two numbers in a row of boxes like so:
Then we generate the next number by adding those first two boxes together.
We continue in this fashion to complete the row:
On the other hand, suppose I give you this:
What numbers should be in the boxes between the 5 and the 16?
Some follow up questions:
- A simpler case: If there are only 3 boxes, and I give you the first and the last boxes, what goes in the middle?
- What happens if I switch the first and last number?
- Can you give a general rule for finding the numbers between the first and last box when there are 5 boxes?
- What if you had 4 boxes? Or 6? Or n? Can you come up with a general rule for finding the numbers in between when you have any number of boxes?
- If the first and last numbers are whole numbers, will the numbers in between be whole numbers? If not, under what conditions will those numbers be whole numbers?
- How can you change the rules? What questions could you ask then?