Types of Recursion The use of recursion goes back to the 19th century. Dedekind  used the notion to obtain functions needed in his formal analysis of the concept of natural number. In logic, recursion appears in Skolem , where it is noted that many basic functions can be defined by simple applications of the method. She presented a paper on the recursive functions at the International Congress of Mathematicians in Zurich in
This stuff is good for something! So how exactly does Linear Algebra relate to the Fibonacci sequence? Now if you have happened to work with the Fibonacci sequence, you should recall that it shows up in some rather odd ways in nature as well as produces some mathematically interesting properties.
A standard model where it is used examines the growth rate of a rabbit population in terms of pairs. If a pair of rabbits is born to each pair of rabbits at the end of each month but it takes each pair two months from birth to give birth for the first time, we can see that at the end of month 1 we have 1 pair.
At the end of month 2, we have still have 2 pairs but at the end of month 3, we suddenly have 2 pairs of rabbits. In month 4, the original pair births their second pair giving 3 pairs of rabbits. At the end of month 5 however, the second pair born has given birth as well as the original so we now have 5 pairs altogether.
The sequence of rabbit pairs that we have over time is then and if we continued on then you would obtain the Fibonacci sequence.
Although this is a rather artificial case, we see the Fibonacci sequence in other ways too, such as in the arrangement of certain flower petals or even in some pine cones. Since the purpose of this post is to talk about linear algebra in the Fibonacci sequence and not the Fibonacci sequence per se, I will leave you to research further applications of the sequence and we will proceed to how it connects with Linear Algebra.
To be precise we should start with the actual definition of the sequence: The Fibonacci Sequence satisfies the recursive relationship where. Actually, the Fibonacci sequence satisfies the relationship given for only some initial conditions where but I could choose to let be anything.
Thus, the Fibonacci sequence we are familiar with is really just a subset of sequences that satisfy our initial relationship.
Similarly, the use of linear algebra is not restricted to this particular case but can be expanded into other recursive relationships. Now suppose that we want to calculate the 18th Fibonacci number.
You can do this several ways. The brute force method could be used to find the numbers by getting out your calculator and start adding numbers. You can also find an equation that gives a closed form solution to the Fibonacci sequence i. We can also use linear algebra to do the same task and it gives us a beautiful result.
If I set up the linear system then I can represent this in the matrix equation:watch my lecture videos, solve some problems, and enjoy the wonders of the Fibonacci sequence and the golden ratio.
Jeffrey R. Chasnov Hong Kong Fibonacci’s rabbit population consists of juveniles and adults. we can use a trick and write F = 1 + 1 F, which is a recursive deﬁnition that can be continued as F = 1 + 1 1 + 1 F, and so.
More than one way to skin a rabbit; the Fibonacci sequence revisited. David H. Ahl.
Many moons ago, we printed a letter from Konrad Kossman with a longish program for producing the Fibonacci sequence.
The Fibonacci sequence is a recursive series of numbers following the rule that any number is the sum of the previous two. Since starting with 0 would result in an unending s eries of zeros. Example 3 Write an equation for the nth term of the arithmetic sequence 5, 15, 25, 35, The first term is 5, and the common difference is a n = a A rabbit population over 8 consecutive years is given by 50, 80, , , , , Chapter 6 Exponential Functions and Sequences.
Such a definition is said to be a recursive definition, and the formula is called Because of its source in Fibonacci's rabbit problem, the sequence 1' 1' 2, 3, 5, 8, 13,..• is called the Fibonacci sequence, and its terms are called Fibonacci numbers. We shall denote the nth Fibonacci number by Fn; thus.
Problem H n is a Fibonacci number if and only if 5n 2 +4 or 5n is a square posed and solved by I Gessel in Fibonacci Quarterly () vol 10, page The method above needs to square the number n being tested and then has to check the new number 5 n 2 ± 4 is a square number.