The Fibonacci numbers are a well-known recursive progression. Most students are introduced to these numbers via the pattern \(1, 1, 2, 3, 5, 8, 13, 21, \ldots\) and are told that the next Fibonacci number is the sum of the previous two. The recursive formula is $$F_{n} = F_{n-1} + F_{n-2}$$ where \(F_{1} = F_{2} = 1\) so that the sequence may begin.

Students are also told fun factoids about the golden ratio and its use in architecture, art, etc. and its relation to Fibonacci numbers. The golden ratio is defined as a limit of successive terms of the Fibonacci sequence. Specifically, let \(\phi\) be the golden ratio, then \(\phi\) is defined as $$\phi = \lim_{n\rightarrow\infty}\frac{F_{n}}{F_{n-1}} = \frac{1 + \sqrt{5}}{2}$$ Since most students are not exposed to this formality when the golden ratio is first mentioned (because of the formality surrounding the mathematical symbolism and students tend to be mathematically young), they are asked to compute \(\frac{F_{n}}{F_{n-1}}\) for increasing values of \(n\) to demonstrate this convergence. The convergence is pretty convincing, empirically, since by \(n=15\) the ratio appears to be accurate to four decimal places (\(\approx 1.6180\)).

But why stop here? There are other fun little calculations to do. Why? It’s just fun to do for the sake of doing it. Maybe it will also make students a bit more curious about what other mysteries exist in sequences of numbers. The Fibonacci numbers aren’t just a one-off piece of math trivia. If students are going to be introduced to this sequence and asked to do a few computations, then why not have them do a few more computations to demonstrate that there’s a bit more depth.

So, without much ado, here are a few more fun computations for a light day in the math classroom:

• Have students compute \(F_{n}^{2} – F_{n-1}F_{n+1}\) starting at \(n=2\). What pattern do they notice? Can you get your students to articulate the alternating behavior? Can you get them to make a general statement? Formally, the relationship is \(F_{n}^{2}-F_{n-1}F_{n+1} = (-1)^{n-1}\).
• Have students compute \(F_{1} + F_{3}\) and have them compare it to \(F_{4}\). Next have students compute \(F_{1} + F_{3} + F_{5}\) and have them compare it to \(F_{6}\). Next have them compute \(F_{1} + F_{3} + F_{5} + F_{7}\) and have them compare it to \(F_{8}\). What do they notice? Can they articulate the general pattern? They should be able to articulate something that says \(F_{1} + F_{3} + F_{5} + \cdots + F_{2n-1} = F_{2n}\)
• As in the previous exercise, but now have them consider sums of the even-indexed Fibonacci numbers. Can they come up with the relationship that \(F_{2} + F_{4} + F_{6} + \ldots + F_{2n} = F_{2n+1} – 1\)?