Patterns in the Fibonacci numbers

This special discussion was led by Zach Teitler and concerned patterns in the famous Fibonacci sequence of numbers. This sequence begins with 1,1 and then each successive number is obtained by taking the sum of the previous two. Here are the next few in the list:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169

Which Fibonacci numbers are even? Which ones are odd? Which Fibonacci numbers are divisible by 3? In this discussion we answered these questions and many more.

Here were the core problems we discussed:

  • Which Fibonacci numbers are divisible by 2? By 3? By 5?
  • What patterns do you notice?
  • Next try the sequence $2^n-1$. This one begins 1, 3, 7, 15, 31, 63, 127, 255, 511, …. Which of these numbers are divisible by 3? By 7? By 15?
  • How does this pattern compare with the patterns in the fibonacci numbers? Can you explain why the pattern works?

We also worked on some fun graphical problems:

  • We have some bricks that are 1 unit by 2 units. We want to make a wall 2 units high, and n units wide. How many ways can it be done? The bricks can be vertical or horizontal.
    Screen Shot 2015-03-09 at 12.02.02 AM
  • Now you have a row of n chairs and you want to put students in some of the chairs. The students are taking an exam so they are not allowed to sit right next to each other. How many ways are there to do this?
    Screen Shot 2015-03-09 at 12.02.16 AM

Finally we looked at some more fun patterns:

  • Look at the “running totals” of the Fibonacci numbers. Here are the first few:
    1+1 = 2
    1+1+2 = 4
    1+1+2+3 = 7
    1+1+2+3+5 = 12

    Do you see any patterns in this new sequence? Can you explain it?
  • What if you only add up every other Fibonacci number
    1+2 =
    1+2+5 =
    1+2+5+13 =
    1+2+5+13+34 =

    What do you get? Can you explain this pattern?
  • Look at the “diagonals” in Pascal’s triangle:
    Screen Shot 2015-03-09 at 12.02.25 AM
    Can you explain this pattern?

Download the handout here.

Our work

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