Essay 3: Working with Power Sequences

One of the things that I love most about mathematics is looking at sequences of numbers and seeing patterns that might not be immediately obvious. For example, the sequence 0, 1, 4, 9, 16, 25.....has always been one of great interest to me. Obviously, this is the sequence of square numbers. For me, the interest is in the fact that the difference betweem successive terms creates another fascinating sequence unto itself. By that, I mean a(n) - a(n-1) creates the sequence of odd integers 1, 3, 5, 7, 9, .....We could go a step further and iterate that same algorithm to see that the difference between odd numbers yields the sequence 2, 2, 2, 2, 2, .....Doing this simple process started me thinking along many different lines. Here are two particular lines of thought that I followed:

Investigation 1: Deriving and proving the general formula for the sequence of cubes

Investigation 2: Excel explorations (my personal favorite)