Sample Mathematics Proposal

Sample Mathematics Proposal

Field(s) of study or investigation: Mathematics

Describe in a few sentences what you want to do in your independent study or project.

I want to study variants of the Fibonacci numbers and their properties. By tweaking the rules used to generate them, I hope to define new sequences of numbers and see if they have interesting versions of the elegant properties belonging to Fibonacci numbers.

How did you become interested in this topic?

I watched an online video about the Fibonacci sequence and the golden ratio, which showed how they appear everywhere in nature, from the arrangement of seeds in a sunflower to the curve of a spiral shell. But even more intriguing to me were numeric patterns like how the ratio of consecutive terms approaches the golden ratio as we go down the sequence, and how the sum of squares of the first N Fibonacci numbers is equal to the Nth Fibonacci number times the one after that.

We produce Fibonacci numbers by starting with 1 and 1 then continually adding the last two numbers to produce the next, to get 2, 3, 5 and so on. Upon reading more about Fibonacci numbers, I learned that one could define new sequences by starting from different numbers but adding them up in the same way. In this way, the ratio of consecutive terms also approaches the golden ratio. I was very intrigued by this, and I'm wondering if I could define sequences similar to the Fibonacci numbers that might have really nice properties like the Fibonacci numbers.

What past experience do you have in this subject area?

I've taken Algebra 1, and I'm comfortable with working with sequences of numbers. We have learnt how to write proofs in class. I learnt a bit of computer programming from a summer camp last year. I also like to watch videos and read books on popular math.

What steps might you take to help you reach your goal?

I'll have to do research on Fibonacci numbers to learn about their properties and their mathematical proofs. For that I can search online or in libraries. I might need to learn new kinds of math, such as solving recurrence relations, that will help this project.

I'll need to define new number sequences using variations on the Fibonacci rules. I'll then need to guess some of their properties, such as any formulas they obey. I might be able to write some computer programs to test many numbers to see if I can identify any pattern. Hopefully I will be able to eventually prove some of the properties I guessed, or prove them wrong.

What kind of guidance do you seek from a mentor?

I'm not familiar with some of the math related to Fibonacci numbers, like "generating functions" and "matrices", which aren't taught in school. I'd like my mentor to point me to some math that might be useful to my project, but which I could learn with guidance from my mentor.

I'm not sure what new rules for producing sequences will be good, or what properties I should look for; I would like some guidance on that. As I am still relatively new to programming and writing proofs, I may need some help when I get stuck. It would also be helpful if my mentor could refine our work plan together so that the difficulty of the work is reasonable and is paced well in a timeline.

What are some challenges that may arise as you carry out this independent study or project? How might you mitigate these challenges?

I might not get very interesting sequences; the new ones I define might not have any nice properties. In that case I would try to imitate how people have analyzed Fibonacci numbers, to borrow their arguments and see if I can get anything interesting from it. If that fails I can try to define new sequences that hopefully obey nicer formulas or are interesting in other ways.

I might also guess some properties that turn out to be difficult to prove, even after trying for very long. I might then try to gather evidence by writing a program to test whether the property is obeyed for many terms in the sequence. Or I might try to show that the property is broken sometimes. At the very least, a guess that has a lot of evidence (program data) behind it is valuable and interesting, even if I can't prove it.