Fibonacci Sequences*

In Secrets of Mental Math, Arthur Benjamin shows how he calculates in his head incredibly fast. I saw Benjamin perform his ‘mathemagical’ tricks once at a math conference. One of his tricks was to ask for two numbers at random from the audience, those were the first two numbers in a Fibonacci sequence. Add them for the third term, add the 2nd and 3rd for the 4th, etc. He said to go ahead and use our calculators, but he told us each new number before we could tell him. We did that for 10 numbers, then he asked us to add them all up. He told us the result so fast, I was amazed.

Although he showed us how he did it, I stayed impressed. His method goes with problem number 3 in Set II (3rd edition, or problems 3 and 4 in Set I in the 1st edition). If you work through it a few times with different sequences, you could get good enough to impress people with your magic trick. (And you wouldn’t have to be a superfast mental calculator, like Benjamin is.)

Figuring out why this works uses a recursive process. Wikipedia definition: “Recursion is … a process of repeating objects in a self-similar way. For instance, when the surfaces of two mirrors are almost parallel with each other the nested images that occur are a form of recursion.” I loved all the patterns in this lesson. I could do some serious math if I tried to see why each one works. I think all the proofs would be recursive.

Here’s a way to see why these Fibonacci sequences are different than the other sequences we’ve studied: The patterns we looked at before all had a way of describing the 20th term, for example, without referring to the other terms. In an arithmetic sequence, the 20th term can be found by adding 19 times the common difference to the first term. In a geometric sequence, you’d multiply the first term by the common ratio to the 19th power. But in a Fibonacci sequence the only way to find the 20th term is to start at the beginning. (The 20th term is the 18th term plus the 19th term, but to find those, you need to find previous terms.) That’s what makes its definition recursive.

I’m afraid I might be over people’s heads on this one. If I am, don’t worry. It just means this topic can be approached in lots of ways, and I’m off in mathland playing... :) While I was working the problems, I wondered if this lesson was harder than the others. If you’re having any trouble with problems, please ask!

And if you enjoyed seeing these patterns, I’ll bet you can find some more cool Fibonacci stuff online.

In Set II, when I was working on problem 8, it was easier for me to think about when I used a way of describing the terms that math folks use. The first term is F1, the second is F2, etc. (Except that the 1 and 2 are supposed to be written as subscripts, which I can’t easily do online yet.) I wrote: sum of squares of first 6 Fibonacci numbers = F6 * F7, so the sum of the squares of the first 10 Fibonacci numbers = … I used that notation to help me think about problems 9 and 10, too.

That’s it for this one. The book has some great extra problems at the end of this chapter, which may take you a while.

Have fun!

-Notes by Sue VanHattum

* The Fibonacci sequence is 1,1,2,3,5,8,... But we can call any sequence with the same rule a Fibonacci sequence: a,b,a+b,b+(a+b),...