Can you Prove it… combinatorially?

This year’s Prove it! Math Academy was a big success, and it was an enormous pleasure to teach the seventeen talented high school students that attended this year. Some of the students mentioned that they felt even more inspired to study math further after our two-week program, but the inspiration went both ways – they inspired me with new ideas as well!

Maria Gemstones - 02

One of the many, many things we investigated at the camp was the Fibonacci sequence, formed by starting with the two numbers $0$ and $1$ and then at each step, adding the previous two numbers to form the next: $$0,1,1,2,3,5,8,13,21,\ldots$$ If $F_n$ denotes the $(n+1)$st term of this sequence (where $F_0=0$ and $F_1=1$), then there is a remarkable formula for the $n$th term, known as Binet’s Formula:

$$F_n=\frac{1}{\sqrt{5}}\left( \left(\frac{1+\sqrt{5}}{2}\right)^n – \left(\frac{1-\sqrt{5}}{2}\right)^n \right)$$

Looks crazy, right? Why would there be $\sqrt 5$’s showing up in a sequence of integers?

At Prove it!, we first derived the formula using generating functions. I mentioned during class that it can also be proven by induction, and later, one of our students was trying to work out the induction proof on a white board outside the classroom. She was amazed how many different proofs there could be of the same fact, and it got me thinking: what if we expand each of the terms using the binomial theorem? Is there a combinatorial proof of the resulting identity?

In particular, suppose we use the binomial theorem to expand $(1+\sqrt{5})^n$ and $(1-\sqrt{5})^n$ in Binet’s formula. The resulting expression is:

$$F_n=\frac{1}{\sqrt{5}\cdot 2^n}\left( \left(\sum_{i=0}^n \binom{n}{i}(\sqrt{5})^i \right) – \left(\sum_{i=0}^n (-1)^i\binom{n}{i}(\sqrt{5})^i \right) \right)$$

Note that the even terms in the two summations cancel, and combining the odd terms gives us:

$$F_n=\frac{1}{\sqrt{5}\cdot 2^n}\left( \sum_{k=0}^{\lfloor n/2\rfloor} 2 \binom{n}{2k+1}(\sqrt{5})^{2k+1} \right)$$
Since $(\sqrt{5})^{2k+1}=\sqrt{5}\cdot 5^k$, we can cancel the factors of $\sqrt{5}$ and multiply both sides by $2^{n-1}$ to obtain:

$$2^{n-1}\cdot F_n=\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k+1}\cdot 5^k.$$

Now, the left hand and right hand side are clearly nonnegative integers, and one handy fact about nonnegative integers is that they count the number of elements in some collection. The proof method of counting in two ways is the simple principle that if by some method one can show that a collection $A$ has $n$ elements, and by another method one can show that $A$ has $m$ elements, then it follows that $m=n$. Such a “combinatorial proof” may be able to be used to prove the identity above, with $m$ being the left hand side of the equation and $n$ being the right.

I started thinking about this after Prove it! ended, and remembered that the $(n+1)$st Fibonacci number $F_n$ counts the number of ways to color a row of $n-2$ fenceposts either black or white such that no two adjacent ones are black. (Can you see why this combinatorial construction would satisfy the Fibonacci recurrence?) For instance, we have $F_5=5$, because there are five such colorings of a row of $3$ fenceposts:
$$\begin{array}{ccc} \square & \square & \square
\\ \\ \square & \square & \blacksquare
\\ \\ \square & \blacksquare & \square
\\ \\ \blacksquare & \square & \square
\\ \\ \blacksquare & \square & \blacksquare

Note also that $2^{n-1}$ counts the number of length-$(n-1)$ sequences of $0$’s and $1$’s. Thus, the left hand side of our identity, $2^{n-1}\cdot F_n$, counts the number of ways of choosing a binary sequence of length $n-1$ and also a fence post coloring of length $n-2$. Because of their lengths, given such a pair we can interlace their entries, forming an alternating sequence of digits and fence posts such as: $$1\, \square\, 0\, \square\, 1\, \blacksquare\, 1$$ We will call such sequences interlaced sequences.

We now need only to show that the right hand side also counts these interlaced sequences. See the next page for my solution, or post your own solution in the comments below!

4 thoughts on “Can you Prove it… combinatorially?

    • Thanks Steven, glad you enjoyed it! 🙂 I wonder if it has been published anywhere in a paper or book – I haven’t found it but maybe I just haven’t looked hard enough. Let me know if you (or anyone else reading this) find it elsewhere.

  1. Hi Maria, pretty cool proof! I have never seen it before.
    Is it OK if we turn this proof into a math problem in our state competition in Brazil?
    Of course, we will credit the proof as yours (or as your camp’s, you choose).

    • Hey Carlos, I think it would be great to turn this into a Brazillian Olympiad problem! (Bearing in mind that the proof is online now, so it should be sufficiently disguised to make it a fair question.) And it can certainly be used as a practice problem. I’ll discuss with the other Prove it! staff to confirm that this is ok and see what they think about the accreditation, and I’ll get back to you privately later.

      Best wishes,

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