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Illustrating and Discovering Some Formulas Geometrically





We have proved formulas for tex2html_wrap_inline790 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 tex2html_wrap_inline796 square into tex2html_wrap_inline798 smaller squares in the natural way. Now color these smaller squares to justify the following formula geometrically:


    Use this to derive a formula for tex2html_wrap_inline800 .

  2. Dissect an tex2html_wrap_inline802 cube into tex2html_wrap_inline804 small cubes in the natural way. Try to color these smaller cubes to get a formula relating tex2html_wrap_inline806 , tex2html_wrap_inline800 , and tex2html_wrap_inline810 . Use this to derive a formula for tex2html_wrap_inline810 .
  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 tex2html_wrap_inline814 , tex2html_wrap_inline816 , and tex2html_wrap_inline818 .

Carl Lee
Wed Jan 6 11:37:02 EST 1999