A Chain of Problems 1

At the moment I'm teaching a Calculus 2 class. In my class I have been giving a sequence of "Toy Problems" that are easy to understand but take a little bit of thinking for a student to solve it.

I'm going to post the problems I gave my students up here for a bit of record keeping. That way I don't have to find them again when I teach a new class. Also I really wanted to try out an online LaTeX compiler I bumped into. It takes TeX and turns it into PNG images. You can find it here: http://hausheer.osola.com/latex2png

The first problems I gave related to Pascal's Triangle. Most of the students have seen it once before in a Precalculus or Algebra class, which makes it a good place to start.

In case you have been on a desert island (or don't know a lot of math) I'll present it to you fresh. Pascal's Triangle has an easy construction:

The top row has only the number 1. The second row has 1 and 1. The third has 1, 2, and 1. You can make each entry in every row by adding the two numbers above it. Notice the 3's in the third row are the sum of 1 and 2 from the second row.

Why would you need it? The most useful application of Pascal's Triangle for students is the Binomial Theorem.

Essentially, you can expand by reading the coefficients off of the n'th row. For instance can been seen to correspond to the 1, 2, 1 in the second row. Of course we can find that one just by using the FOIL method. The third row gives us: .

There are a few more properties of Pascal's Triangle. This leads to the three following problems:

1) Prove that for each node on the triangle the number of ways you can descend from the top of Pascal's Triangle to the node is exactly the number at that node.

2) Prove that the sum of all the numbers on row n (the top is row 0) is .

3) Prove that if a row is prime numbered (a prime is 2, 3, 5, 7, 11, etc.) then that prime divides all of the numbers in the row except the 1's at the ends.