[MUSIC] Let me wrap up this course by briefly reviewing what we've learned. I introduced you to the famous Fibonacci sequence, 1, 1, 2, 3, 5, 8, etc., where each Fibonacci number is the sum of the previous two Fibonacci numbers. For the Fibonacci sequence we start with 1, 1 and then continue the sequence from there. We can start with 1, 3 instead, and then we get the Lucas sequence, 1, 3, 4, 7, 11, etc. I hope you enjoyed learning about both the Fibonacci and Lucas sequences in this course, and the formulas that relate them. You had to do some homework problems with them. I also showed you where the golden ratio comes from. You take a line segment, you cut it into two pieces, one long and one short. And if the ratio of the long piece to the short piece is the same as the ratio of the whole line to long piece, and we say that the long piece and the short piece are in the golden ratio. The golden ratio is equal to the number square root of 5 + 1 divided by 2. It's approximately 1.618, and it's an irrational number. A related number is the golden ratio conjugate. That's equal to square root of 5- 1 over 2. And you can see that the golden ratio conjugate is just the fractional part of the golden ratio. It's also equal to the reciprocal of the golden ratio. The Fibonacci numbers and the golden ratio and the golden ratio conjugate are intimately related. There's a famous formula that we derived in the course called Binet's formula that gives this relationship. Binet's formula lets you calculate the nth Fibonacci number from the nth powers of the golden ratio and the golden ratio conjugate. As you can see, it's a very simple formula, yet I hope you find it as I do to be quite beautiful. I also showed you how the ratio of consecutive Fibonacci numbers can form a sequence of converging, rational approximations to the golden ratio and how the sequence converges very slowly. Because of this slow conversions, we say that the golden ratio is a number that's very difficult to approximate by a rational number. Taking a little poetic license, we can say that the golden ratio is the most irrational of the irrational numbers. We also learn some other interesting mathematics in this course like the Fibonacci Q-Matrix. Here the Q-Matrix is raised to the nth power and expressed as a two by two matrix of Fibonacci numbers. We also derived Cassini's identity. This neat little formula can be interpreted as the difference between the area of a rectangle and the area of a square. And its application is a funny paradoxical puzzle called the Fibonacci bamboozlement. I also showed you how to add the first n Fibonacci numbers and the first n Fibonacci numbers squared. And I showed you how to draw both a golden spiral and a Fibonacci spiral. These two beautiful spirals being very closely related. Finally we saw how the Fibonacci numbers can show up unexpectedly in nature. This, of course, has to do with the specialness of the golden ratio and its relationship to the ratio of consecutive Fibonacci numbers. There's much more material about the Fibonacci numbers and the golden ratio on the web and in a university library. Our library has a lot of material about it, and you might have some fun looking at that. And if you like this math course and you haven't had much chance to study university mathematics, you can look at some of the other MOOC courses on mathematics. I hope this course has shown you that math can be interesting and fun. And I hope you have enjoyed this course. I've certainly had a lot of fun making it. Cheers.