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?

- 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 .

- 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 .
- Try to guess a generalization of the relationship you discovered from the colorings of the square and the cube.
- Consider the following list of equations:
Sum these equations and use this to prove that

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

Wed Jan 6 11:37:02 EST 1999