Illustrating and Discovering Some Formulas Geometrically

Define

and

We have proved formulas for when k is small, but what about finding formulas for higher values of k? Is there any systematic way to do this?

1. Dissect an square into smaller squares in the natural way. Now color these smaller squares to justify the following formula geometrically:

   

   Use this to derive a formula for .

2. Dissect an cube into small cubes in the natural way. Try to color these smaller cubes to get a formula relating , , and . Use this to derive a formula for .
3. Try to guess a generalization of the relationship you discovered from the colorings of the square and the cube.
4. Consider the following list of equations:

   

   Sum these equations and use this to prove that

   

5. Use the above to find formulas for , , and .