Pascal's other Discoveries

Now that we understand how the triangle works, let's take a look at some of the amazing patterns that can be seen within the triangle. First, notice that the triangle is symmetrical. Another easy pattern to see is the patterns of the diagonals; the first row are ones, the next are counting numbers, the third are triangular numbers, and so on.  Then, if we sum the rows, they double each row. It is also possible to find patterns with the prime numbers and the Fibonacci numbers as well. A useful pattern that we often see  and apply is that you can use the values in a given row to show you the coefficients when expanding polynomials. Let's see this in an example: