## The quadratic formula and Fibonacci numbers

I knew ahead of time that I was going to have a busy week of work this week and was looking for something fun to cover with my older son in the limited amount of time that we would have.  We were supposed to be covering properties of functions so I was looking for a topic that would at least be tangentally related to that, but I also wanted to get him a little review with solving quadratic equations.  Diving into the formula for the Fibonacci numbers seemed to fit the bill quite nicely . . .

We started with a short talk about the Fibonacci numbers that focused on thinking about the usual recurrence relation definition:

I didn’t do a great job with the graph at the end, and we spent another 10 minutes after the video talking through the graph and comparing it to $y = x^2$ and $y = 2^x.$  I was pretty happy with how that talk about the graphs went after the first video and wanted to reinforce some of those ideas the next day.  With that in mind, the next talk begins by discussing how the Fibonacci numbers grow and then considers what happens if the Fibonacci numbers could be written in the form $F(n) = C * \lambda^n$.

I’ve always found this technique for finding the closed form for the Fibonacci numbers to be really beautiful.  Turns out it is a great little tool for algebra review, too!

After finding that the Fibonacci numbers somehow related to the numbers $\frac{1 \pm \sqrt{5}}{2}$ at the end of the last video, in this one we finish up the calculation and write down the closed form for each Fibonacci number.  Lots of great algebra review for kids in these calculations, too!

With the formula for the Fibonacci numbers now in hand, I wanted to play around with the formula so we jumped over to Wolfram Alpha.  The first neat thing I wanted to go through was how we could now easily calculate each Fibonacci number without reference to the prior two numbers.  Fun!  The second thing I wanted to show was that the second term in the formula doesn’t play much of a role for the larger Fibonacci numbers.  It is pretty amazing to see how well $\frac{1}{\sqrt{5}} * (\frac{ 1 + \sqrt{5}}{2})^n$ approximates the larger Fibonacci numbers.  The meat of the formula is all in the first term!  I thought this would be an especially fun fact to show him since the two terms look so similar when you write them down.

We’ve talked a little bit about the Fibonacci numbers previously, but not with this level of math.  I’d chosen this topic because I thought the math is really interesting – which it was – but all of the algebra review turned this in to a nice learning opportunity, too.  For what was supposed to be just a  little diversion way from the book during a busy week of work for me, this topic turned out to be really fun.