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be read or edited using a copy of Mathematica or MathReader. If you
received this as email, use your mail application or copy/paste to save
everything from the line containing (*^ down to the line containing ^*)
into a plain text file. On some systems you may have to give the file a
name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
The line below identifies what version of Mathematica created this file,
but it can be opened using any other version as well.";
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A GEOMETRIC DISCOVERY
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The Sum of the Squares of the First n Integers
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Introduction
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In beginning calculus it is common to show how to evaluate #¼õxÛdx directly from the definition of the definite integral by using the formula [n(n+1)(2n+1)]/6 for the sum of the squares of the first n integers, 1Û+2Û+3Û+...+nÛ. In most texts this formula is presented without consideration of its genesis. The formula is usually proved using mathematical induction, but induction really does little to explain how the formula was discovered in the first place; in fact, students often object that there is a certain "circularity" to the process since they are being asked to prove something that they have no real reason to predict might be true. This project shows that there is a simple geometric technique for demonstrating the origin of the formula. This technique can even be used by pre-algebra students to find the sum of the first n squares for any particular n even if they have not yet learned to use a variable. For students who do understand variables, working with the this technique and Mathematica can provide a nice exercise in algebraic identities.
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The Basic Geometric Model of the Sum
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Mathematica can display a three dimensional model of the sum 1Û+2Û+3Û+4Û using unit cubes.
;[s]
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Show[cL,Boxed->False,
Lighting->False,
ViewPoint->{1.300, -2.400, 2.000}];
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The top cube in the figure "SumSquares" represents 1Û, the next layer of four cubes represents 2Û, the third layer of nine cubes represents 3Û, and the bottom layer of sixteen cubes represents 4Û.
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The Sum Embedded in a Cube
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A cube 5 units on an edge can be thought of as the sum of three of these "pyramid" models of the basic sum together with several other unit cubes which serve to separate the three "pyramids" and complete the larger cube. The figure "BigCube" shows the three basic sum "pyramids" in light gray in the lower rear left, the lower rear right, and the top front corners. The darker unit cubes separate the three basic sum "pyramids."
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View of the Three Basic Sum "Pyramids."
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The figure "3Pyramids" shows the three basic sum pyramids moved apart slightly and without the separating unit cubes.
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c2L = Graphics3D[{
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c3L = Graphics3D[{
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c4L = Graphics3D[
Text[FontForm[
"3Pyramids",
{"Chicago",10}],{5,5,0}]];
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Clear[pc1L,pc2L,pc3L,pc4L]
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pc1L = Show[c1L,Boxed->False,
Lighting->False,
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DisplayFunction->Identity];
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pc2L = Show[c2L,Boxed->False,
Lighting->False,
ViewPoint->{1.500, -2.500, 2.500},
DisplayFunction->Identity];
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pc3L = Show[c3L,Boxed->False,
Lighting->False,
ViewPoint->{1.500, -1.500, 1.300},
DisplayFunction->Identity];
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pc4L = Show[c4L,Boxed->False,
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DisplayFunction->Identity];
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Show[Graphics[{
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View of the Three Basic Sum "Pyramids" with the Separating Cubes
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The figure "3SeparatedPyramids" shows the three basic sum "pyramids" in light gray and the separating unit cubes in medium and dark gray.
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Clear[c1,c2,c3,c4,c5,c6,c7,c8]
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c1 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,
Flatten[
Table[
Table[
Table[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],
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c2 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[
Table[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];
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c3 = Graphics3D[{Text[FontForm[
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Table[
Table[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];
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c4 = Graphics3D[{
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Flatten[
Table[
Table[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],
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c5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],
2]]}];
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c6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,3}],
2]]}];
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c7 = Graphics3D[{GrayLevel[.25],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],
2]]}];
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p41 = Show[c1,
Lighting->False,
Boxed->False,
ViewPoint->{2.500, -1.500, 1.300},
DisplayFunction->Identity];
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p42 = Show[c2,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -2.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p43 = Show[c3,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -1.500, -1.400},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p44 = Show[c4,c5,c6,c7,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -1.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect; endGroup; endGroup; endGroup]
Show[Graphics[{
Rectangle[Scaled[{0,0}],Scaled[{.5,.5}],p41],
Rectangle[Scaled[{.5,0}],Scaled[{1,.5}],p42],
Rectangle[Scaled[{.25,.55}],Scaled[{.75,1}],p43],
Rectangle[Scaled[{.25,.12}],Scaled[{.75,.62}],p44]
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DisplayFunction->$DisplayFunction];
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Completely Separated
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The figure "AllSeparated" shows the 3 basic sum "pyramids" in light gray, the unit cubes which are the interfaces separating two "pyramids" in medium gray, and the main diagonal of 5 unit cubes in dark gray.
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Clear[c1,c2,c3,c4,c5,c6,c7]
:[font = input; preserveAspect]
c1 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,
Flatten[
Table[
Table[
Table[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],
2]]}];
:[font = input; preserveAspect]
c2 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[
Table[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];
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c3 = Graphics3D[{Text[FontForm[
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{"Chicago",10}],{5,0,6}],
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];
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c4 = Graphics3D[{
GrayLevel[0.5],Map[Cuboid,
Flatten[
Table[
Table[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],
2]]}];
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c5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],
2]]}];
:[font = input; preserveAspect]
c6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,4}],
2]]}];
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c7 = Graphics3D[{GrayLevel[.25],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],
2]]}];
:[font = input; preserveAspect]
Clear[p71,p72,p73,p74,p75,p76,p77]
:[font = input; preserveAspect]
p71 = Show[c1,
Lighting->False,
Boxed->False,
ViewPoint->{2.500, -1.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p72 = Show[c2,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -2.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p73 = Show[c3,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -1.500, -1.400},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p74 = Show[c4,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -1.500, 1.300},
DisplayFunction->Identity];
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p75 = Show[c5,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -1.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p76 = Show[c6,
Lighting->False,
Boxed->False,
ViewPoint->{1.500, -2.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect]
p77 = Show[c7,
Lighting->False,
Boxed->False,
ViewPoint->{2.500, -1.500, 1.300},
DisplayFunction->Identity];
:[font = input; preserveAspect; endGroup; endGroup; endGroup]
Show[Graphics[{
Rectangle[Scaled[{0,0}],Scaled[{.5,.5}],p71],
Rectangle[Scaled[{.5,0}],Scaled[{1,.5}],p72],
Rectangle[Scaled[{.25,.55}],Scaled[{.75,1}],p73],
Rectangle[Scaled[{0,.5}],Scaled[{.5,1}],p74],
Rectangle[Scaled[{.5,.5}],Scaled[{1,1}],p75],
Rectangle[Scaled[{.35,0}],Scaled[{.85,.4}],p76],
Rectangle[Scaled[{.15,0}],Scaled[{.65,.5}],p77]
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DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
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Begin Reassembling the Cube
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The figure "ReJoin(1)" shows some parts of the cube reassembled. The basic sum "pyramids" from the lower rear left and right are shown in light gray, the three sheets of cubes that separate the basic sum "pyramids" are in medium gray, and the five cubes of the main diagonal are in dark gray. The top front "pyramid" is not shown.
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Clear[c1,c2,c3,c4,c5,c6,c7]
:[font = input; preserveAspect]
c1 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[
Table[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],2]]}];
:[font = input; preserveAspect]
c4 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],2]]}];
:[font = input; preserveAspect]
c2 = Graphics3D[{
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[
Table[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];
:[font = input; preserveAspect]
c5 = Graphics3D[{
GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],2]]}];
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c6 = Graphics3D[{
Text[FontForm[
"ReJoin(1)",{"Chicago",10}],{1,0,0}],
GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,3}],2]]}];
:[font = input; preserveAspect]
c7 = Graphics3D[{GrayLevel[0.25],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],2]]}];
:[font = input; preserveAspect]
Clear[r1,r2,r3]
:[font = input; preserveAspect]
r1 = Show[c1,c4,
ViewPoint->{2.500, -1.500, 2.500},
Lighting->False,
Boxed->False,
DisplayFunction->Identity];
:[font = input; preserveAspect]
r2 = Show[c2,c5,
ViewPoint->{1.500, -2.500, 2.500},
Lighting->False,
Boxed->False,
DisplayFunction->Identity];
:[font = input; preserveAspect]
r3 = Show[c6,c7,
Lighting->False,
Boxed->False,
ViewPoint->{1.300, -2.400, 2.000},
DisplayFunction->Identity];
:[font = input; preserveAspect; endGroup; endGroup; endGroup]
Show[Graphics[{
Rectangle[Scaled[{0,.25}],Scaled[{.5,.75}],r1],
Rectangle[Scaled[{.5,.25}],Scaled[{1,.75}],r2],
Rectangle[Scaled[{.25,.25}],Scaled[{.75,.75}],r3]
}],
DisplayFunction->$DisplayFunction];
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Continue Reassembling the Cube
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The figure "ReJoin(2)" shows all of the parts of the cube reassembled except for the top front "pyramid."
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Clear[c1,c2,c3,c4,c5,c6,c7]
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c1 = Graphics3D[{GrayLevel[0.8],Map[Cuboid,Flatten[
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c2 = Graphics3D[{
Text[FontForm["Rejoin(2)",{"Chicago",10}],{7,2,0}],
GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[
Table[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];
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c3 = Graphics3D[{GrayLevel[0.8],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];
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c4 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],2]]}];
:[font = input; preserveAspect]
c5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
Table[
Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],2]]}];
:[font = input; preserveAspect]
c6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[
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The Cube Revisited
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Another look at the figure "BigCube" shows all three of the basic sum pyramids and the separating unit cubes reassembled.
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
p
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P
p
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
p
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
p
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P
p
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P
p
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P
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P
p
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F P
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P
p
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F P
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P
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p
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p
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p
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p
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p
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p
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p
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p
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p
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p
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p
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p
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p
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P
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F P
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F P
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P
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P
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F P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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F P
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P
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P
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P
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P
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P
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P
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P
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P
p
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
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P
p
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P
p
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P
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P
p
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P
p
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P
p
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P
p
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P
p
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P
p
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F P
s
P
p
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.41372 .5021 m .48226 .4637 L .55079 .5021 L p
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F P
s
P
p
.002 w
.48226 .4637 m .55079 .5021 L .55278 .58594 L p
.5 g
F P
s
P
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.002 w
.48226 .54929 m .48226 .4637 L .41372 .5021 L p
.5 g
F P
s
P
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.002 w
.41372 .5021 m .41173 .58594 L .48226 .54929 L p
.5 g
F P
s
P
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.002 w
.48226 .4637 m .55079 .5021 L .55278 .58594 L p
.5 g
F P
s
P
p
.002 w
.27665 .5021 m .34058 .4637 L .41372 .5021 L p
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F P
s
P
p
.002 w
.27665 .5021 m .34058 .4637 L .41372 .5021 L p
.8 g
F P
s
P
p
.002 w
.33633 .54929 m .34058 .4637 L .27665 .5021 L p
.5 g
F P
s
P
p
.002 w
.27665 .5021 m .27069 .58594 L .33633 .54929 L p
.5 g
F P
s
P
p
.002 w
.34058 .4637 m .41372 .5021 L .41173 .58594 L p
.5 g
F P
s
P
p
.002 w
.27665 .5021 m .1989 .4637 L .20692 .38295 L p
.8 g
F P
s
P
p
.002 w
.27665 .5021 m .1989 .4637 L .20692 .38295 L p
.8 g
F P
s
P
p
.002 w
.34058 .4637 m .27665 .5021 L .1989 .4637 L p
.5 g
F P
s
P
p
.002 w
.33633 .54929 m .34058 .4637 L .27665 .5021 L p
.5 g
F P
s
P
p
.002 w
.27665 .5021 m .27069 .58594 L .33633 .54929 L p
.5 g
F P
s
P
p
.002 w
.13958 .5021 m .1989 .4637 L .27665 .5021 L p
.8 g
F P
s
P
p
.002 w
.13958 .5021 m .1989 .4637 L .27665 .5021 L p
.8 g
F P
s
P
p
.002 w
.1989 .4637 m .27665 .5021 L .27069 .58594 L p
.5 g
F P
s
P
p
.002 w
.62818 .54929 m .62393 .4637 L .55079 .5021 L p
.8 g
F P
s
P
p
.002 w
.55079 .5021 m .55278 .58594 L .62818 .54929 L p
.8 g
F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.68786 .5021 m .62393 .4637 L .61992 .38295 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.34058 .4637 m .27665 .5021 L .1989 .4637 L p
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F P
s
P
p
.002 w
.1904 .54929 m .1989 .4637 L .13958 .5021 L p
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F P
s
P
p
.002 w
.13958 .5021 m .12964 .58594 L .1904 .54929 L p
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F P
s
P
p
.002 w
.1989 .4637 m .27665 .5021 L .27069 .58594 L p
.8 g
F P
s
P
p
.002 w
.48226 .4637 m .41372 .5021 L .34058 .4637 L p
.8 g
F P
s
P
p
.002 w
.48226 .54929 m .48226 .4637 L .41372 .5021 L p
.8 g
F P
s
P
p
.002 w
.41372 .5021 m .41173 .58594 L .48226 .54929 L p
.8 g
F P
s
P
p
.002 w
.34058 .4637 m .41372 .5021 L .41173 .58594 L p
.8 g
F P
s
P
p
.002 w
.34459 .38295 m .4111 .34036 L .48226 .38295 L p
.25 g
F P
s
P
p
.002 w
.34459 .38295 m .4111 .34036 L .48226 .38295 L p
.5 g
F P
s
P
p
.002 w
.4111 .34036 m .48226 .38295 L .48226 .4637 L p
.25 g
F P
s
P
p
.002 w
.48226 .38295 m .4111 .34036 L .41312 .26276 L p
.5 g
F P
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P
p
.002 w
.48226 .38295 m .4111 .34036 L .41312 .26276 L p
.8 g
F P
s
P
p
.002 w
.55341 .34036 m .48226 .38295 L .4111 .34036 L p
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F P
s
P
p
.002 w
.48226 .38295 m .55341 .34036 L .61992 .38295 L p
.8 g
F P
s
P
p
.002 w
.48226 .38295 m .55341 .34036 L .61992 .38295 L p
.8 g
F P
s
P
p
.002 w
.55556 .42263 m .55341 .34036 L .48226 .38295 L p
.5 g
F P
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P
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.002 w
.48226 .38295 m .48226 .4637 L .55556 .42263 L p
.5 g
F P
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P
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.002 w
.4111 .34036 m .48226 .38295 L .48226 .4637 L p
.5 g
F P
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P
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.002 w
.55556 .42263 m .55341 .34036 L .48226 .38295 L p
.8 g
F P
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P
p
.002 w
.48226 .38295 m .48226 .4637 L .55556 .42263 L p
.8 g
F P
s
P
p
.002 w
.55341 .34036 m .61992 .38295 L .62393 .4637 L p
.8 g
F P
s
P
p
.002 w
.61992 .38295 m .55341 .34036 L .55139 .26276 L p
.8 g
F P
s
P
p
.002 w
.55341 .34036 m .48226 .38295 L .4111 .34036 L p
.8 g
F P
s
P
p
.002 w
.20692 .38295 m .26878 .34036 L .34459 .38295 L p
.8 g
F P
s
P
p
.002 w
.20692 .38295 m .26878 .34036 L .34459 .38295 L p
.8 g
F P
s
P
p
.002 w
.26235 .42263 m .26878 .34036 L .20692 .38295 L p
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F P
s
P
p
.002 w
.20692 .38295 m .1989 .4637 L .26235 .42263 L p
.8 g
F P
s
P
p
.002 w
.40895 .42263 m .4111 .34036 L .34459 .38295 L p
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F P
s
P
p
.002 w
.34459 .38295 m .34058 .4637 L .40895 .42263 L p
.25 g
F P
s
P
p
.002 w
.34459 .38295 m .26878 .34036 L .27485 .26276 L p
.8 g
F P
s
P
p
.002 w
.34459 .38295 m .26878 .34036 L .27485 .26276 L p
.8 g
F P
s
P
p
.002 w
.4111 .34036 m .34459 .38295 L .26878 .34036 L p
.5 g
F P
s
P
p
.002 w
.40895 .42263 m .4111 .34036 L .34459 .38295 L p
.5 g
F P
s
P
p
.002 w
.34459 .38295 m .34058 .4637 L .40895 .42263 L p
.5 g
F P
s
P
p
.002 w
.26878 .34036 m .34459 .38295 L .34058 .4637 L p
.5 g
F P
s
P
p
.002 w
.4111 .34036 m .34459 .38295 L .26878 .34036 L p
.8 g
F P
s
P
p
.002 w
.26878 .34036 m .34459 .38295 L .34058 .4637 L p
.8 g
F P
s
P
p
.002 w
.48226 .2159 m .41312 .26276 L .3393 .2159 L p
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F P
s
P
p
.002 w
.41312 .26276 m .48226 .2159 L .55139 .26276 L p
.8 g
F P
s
P
p
.002 w
.48226 .29479 m .48226 .2159 L .41312 .26276 L p
.25 g
F P
s
P
p
.002 w
.41312 .26276 m .4111 .34036 L .48226 .29479 L p
.25 g
F P
s
P
p
.002 w
.27485 .26276 m .3393 .2159 L .41312 .26276 L p
.8 g
F P
s
P
p
.002 w
.3393 .2159 m .41312 .26276 L .4111 .34036 L p
.25 g
F P
s
P
p
.002 w
.48226 .29479 m .48226 .2159 L .41312 .26276 L p
.8 g
F P
s
P
p
.002 w
.41312 .26276 m .4111 .34036 L .48226 .29479 L p
.8 g
F P
s
P
p
.002 w
.48226 .2159 m .55139 .26276 L .55341 .34036 L p
.8 g
F P
s
P
p
.002 w
.33497 .29479 m .3393 .2159 L .27485 .26276 L p
.8 g
F P
s
P
p
.002 w
.27485 .26276 m .26878 .34036 L .33497 .29479 L p
.8 g
F P
s
P
p
.002 w
.3393 .2159 m .41312 .26276 L .4111 .34036 L p
.8 g
F P
s
P
p
.002 w
.69195 .82718 m .62685 .79913 L .62268 .7071 L p
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F P
s
P
p
.002 w
.69195 .82718 m .62685 .79913 L .62268 .7071 L p
.5 g
F P
s
P
p
.002 w
.77144 .79913 m .69195 .82718 L .62685 .79913 L p
.5 g
F P
s
P
p
.002 w
.55216 .82718 m .62685 .79913 L .69195 .82718 L p
.5 g
F P
s
P
p
.002 w
.13276 .82718 m .19307 .79913 L .27256 .82718 L p
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F P
s
P
p
.002 w
.27256 .82718 m .19307 .79913 L .20142 .7071 L p
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F P
s
P
p
.002 w
.13276 .82718 m .04848 .79913 L .061 .7071 L p
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F P
s
P
p
.002 w
.19307 .79913 m .13276 .82718 L .04848 .79913 L p
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F P
s
P
p
.002 w
.55216 .82718 m .48226 .79913 L .48226 .7071 L p
.8 g
F P
s
P
p
.002 w
.55216 .82718 m .48226 .79913 L .48226 .7071 L p
.8 g
F P
s
P
p
.002 w
.62685 .79913 m .55216 .82718 L .48226 .79913 L p
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F P
s
P
p
.002 w
.41236 .82718 m .48226 .79913 L .55216 .82718 L p
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F P
s
P
p
.002 w
.41236 .82718 m .33766 .79913 L .34184 .7071 L p
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F P
s
P
p
.002 w
.41236 .82718 m .33766 .79913 L .34184 .7071 L p
.8 g
F P
s
P
p
.002 w
.48226 .79913 m .41236 .82718 L .33766 .79913 L p
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F P
s
P
p
.002 w
.27256 .82718 m .33766 .79913 L .41236 .82718 L p
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F P
s
P
p
.002 w
.27256 .82718 m .19307 .79913 L .20142 .7071 L p
.8 g
F P
s
P
p
.002 w
.33766 .79913 m .27256 .82718 L .19307 .79913 L p
.8 g
F P
s
P
p
.002 w
.62268 .7071 m .70014 .67478 L .76309 .7071 L p
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F P
s
P
p
.002 w
.62268 .7071 m .70014 .67478 L .76309 .7071 L p
.8 g
F P
s
P
p
.002 w
.70685 .7691 m .70014 .67478 L .62268 .7071 L p
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F P
s
P
p
.002 w
.62268 .7071 m .62685 .79913 L .70685 .7691 L p
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F P
s
P
p
.002 w
.70014 .67478 m .76309 .7071 L .77144 .79913 L p
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F P
s
P
p
.002 w
.62268 .7071 m .55488 .67478 L .55278 .58594 L p
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F P
s
P
p
.002 w
.62268 .7071 m .55488 .67478 L .55278 .58594 L p
.5 g
F P
s
P
p
.002 w
.70014 .67478 m .62268 .7071 L .55488 .67478 L p
.5 g
F P
s
P
p
.002 w
.48226 .7071 m .55488 .67478 L .62268 .7071 L p
.5 g
F P
s
P
p
.002 w
.061 .7071 m .11911 .67478 L .20142 .7071 L p
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F P
s
P
p
.002 w
.061 .7071 m .11911 .67478 L .20142 .7071 L p
.8 g
F P
s
P
p
.002 w
.10794 .7691 m .11911 .67478 L .061 .7071 L p
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F P
s
P
p
.002 w
.061 .7071 m .04848 .79913 L .10794 .7691 L p
.5 g
F P
s
P
p
.002 w
.11911 .67478 m .20142 .7071 L .19307 .79913 L p
.5 g
F P
s
P
p
.002 w
.20142 .7071 m .26437 .67478 L .34184 .7071 L p
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F P
s
P
p
.002 w
.34184 .7071 m .26437 .67478 L .27069 .58594 L p
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F P
s
P
p
.002 w
.20142 .7071 m .11911 .67478 L .12964 .58594 L p
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F P
s
P
p
.002 w
.20142 .7071 m .11911 .67478 L .12964 .58594 L p
.8 g
F P
s
P
p
.002 w
.26437 .67478 m .20142 .7071 L .11911 .67478 L p
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F P
s
P
p
.002 w
.76309 .7071 m .70014 .67478 L .69382 .58594 L p
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F P
s
P
p
.002 w
.70014 .67478 m .62268 .7071 L .55488 .67478 L p
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F P
s
P
p
.002 w
.70685 .7691 m .70014 .67478 L .62268 .7071 L p
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F P
s
P
p
.002 w
.62268 .7071 m .62685 .79913 L .70685 .7691 L p
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F P
s
P
p
.002 w
.48226 .7071 m .55488 .67478 L .62268 .7071 L p
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F P
s
P
p
.002 w
.55488 .67478 m .62268 .7071 L .62685 .79913 L p
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F P
s
P
p
.002 w
.55712 .7691 m .55488 .67478 L .48226 .7071 L p
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F P
s
P
p
.002 w
.48226 .7071 m .48226 .79913 L .55712 .7691 L p
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F P
s
P
p
.002 w
.55488 .67478 m .62268 .7071 L .62685 .79913 L p
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F P
s
P
p
.002 w
.48226 .7071 m .40963 .67478 L .41173 .58594 L p
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F P
s
P
p
.002 w
.48226 .7071 m .40963 .67478 L .41173 .58594 L p
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F P
s
P
p
.002 w
.55488 .67478 m .48226 .7071 L .40963 .67478 L p
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F P
s
P
p
.002 w
.55712 .7691 m .55488 .67478 L .48226 .7071 L p
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F P
s
P
p
.002 w
.48226 .7071 m .48226 .79913 L .55712 .7691 L p
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F P
s
P
p
.002 w
.34184 .7071 m .40963 .67478 L .48226 .7071 L p
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F P
s
P
p
.002 w
.34184 .7071 m .40963 .67478 L .48226 .7071 L p
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F P
s
P
p
.002 w
.40963 .67478 m .48226 .7071 L .48226 .79913 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.34184 .7071 m .33766 .79913 L .40739 .7691 L p
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F P
s
P
p
.002 w
.40963 .67478 m .48226 .7071 L .48226 .79913 L p
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F P
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P
p
.002 w
.34184 .7071 m .26437 .67478 L .27069 .58594 L p
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F P
s
P
p
.002 w
.40963 .67478 m .34184 .7071 L .26437 .67478 L p
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F P
s
P
p
.002 w
.40739 .7691 m .40963 .67478 L .34184 .7071 L p
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F P
s
P
p
.002 w
.34184 .7071 m .33766 .79913 L .40739 .7691 L p
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F P
s
P
p
.002 w
.20142 .7071 m .26437 .67478 L .34184 .7071 L p
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F P
s
P
p
.002 w
.26437 .67478 m .34184 .7071 L .33766 .79913 L p
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F P
s
P
p
.002 w
.25767 .7691 m .26437 .67478 L .20142 .7071 L p
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F P
s
P
p
.002 w
.20142 .7071 m .19307 .79913 L .25767 .7691 L p
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F P
s
P
p
.002 w
.26437 .67478 m .34184 .7071 L .33766 .79913 L p
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F P
s
P
p
.002 w
.26437 .67478 m .20142 .7071 L .11911 .67478 L p
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F P
s
P
p
.002 w
.25767 .7691 m .26437 .67478 L .20142 .7071 L p
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F P
s
P
p
.002 w
.20142 .7071 m .19307 .79913 L .25767 .7691 L p
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F P
s
P
p
.002 w
.11911 .67478 m .20142 .7071 L .19307 .79913 L p
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F P
s
P
p
.002 w
.55488 .67478 m .48226 .7071 L .40963 .67478 L p
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F P
s
P
p
.002 w
.40963 .67478 m .34184 .7071 L .26437 .67478 L p
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F P
s
P
p
.002 w
.55278 .58594 m .62818 .54929 L .69382 .58594 L p
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F P
s
P
p
.002 w
.55278 .58594 m .62818 .54929 L .69382 .58594 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.55278 .58594 m .55488 .67478 L .63269 .64017 L p
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F P
s
P
p
.002 w
.62818 .54929 m .69382 .58594 L .70014 .67478 L p
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F P
s
P
p
.002 w
.55278 .58594 m .48226 .54929 L .48226 .4637 L p
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F P
s
P
p
.002 w
.55278 .58594 m .48226 .54929 L .48226 .4637 L p
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F P
s
P
p
.002 w
.62818 .54929 m .55278 .58594 L .48226 .54929 L p
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F P
s
P
p
.002 w
.41173 .58594 m .48226 .54929 L .55278 .58594 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.12964 .58594 m .1904 .54929 L .27069 .58594 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.12964 .58594 m .11911 .67478 L .18138 .64017 L p
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F P
s
P
p
.002 w
.1904 .54929 m .27069 .58594 L .26437 .67478 L p
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F P
s
P
p
.002 w
.27069 .58594 m .33633 .54929 L .41173 .58594 L p
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F P
s
P
p
.002 w
.41173 .58594 m .33633 .54929 L .34058 .4637 L p
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F P
s
P
p
.002 w
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F P
s
P
p
.002 w
.27069 .58594 m .1904 .54929 L .1989 .4637 L p
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F P
s
P
p
.002 w
.33633 .54929 m .27069 .58594 L .1904 .54929 L p
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F P
s
P
p
.002 w
.69382 .58594 m .62818 .54929 L .62393 .4637 L p
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F P
s
P
p
.002 w
.62818 .54929 m .55278 .58594 L .48226 .54929 L p
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F P
s
P
p
.002 w
.63269 .64017 m .62818 .54929 L .55278 .58594 L p
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F P
s
P
p
.002 w
.55278 .58594 m .55488 .67478 L .63269 .64017 L p
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F P
s
P
p
.002 w
.41173 .58594 m .48226 .54929 L .55278 .58594 L p
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F P
s
P
p
.002 w
.48226 .54929 m .55278 .58594 L .55488 .67478 L p
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F P
s
P
p
.002 w
.48226 .64017 m .48226 .54929 L .41173 .58594 L p
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F P
s
P
p
.002 w
.41173 .58594 m .40963 .67478 L .48226 .64017 L p
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F P
s
P
p
.002 w
.48226 .54929 m .55278 .58594 L .55488 .67478 L p
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F P
s
P
p
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/Chicago findfont 10 scalefont setfont
[(BigCube)] .70422 .16575 0 0 Mshowa
P
P
% End of Graphics
MathPictureEnd
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Finding the Sum 1Û+2Û+3Û+...+nÛ
:[font = text; inactive; preserveAspect; endGroup]
Now let's consider how the geometric model leads to the formula for the sum of the first n squares (1Û+2Û+3Û+...+nÛ).
Each of the three light-gray basic sum "pyramids" represents the sum 1Û+2Û+3Û+4Û. The three medium-gray sheets that separate the three basic sum "pyramids" each represent the sum 1+2+3+4.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Small Digression
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
Students studying sums like 1Û+2Û+3Û+4Û are probably already familiar with sums like 1+2+3+4, but, if not, the the figure "IntegerSum," provides a simple two-dimensional analogue to our three-dimensional technique. Clearly 1+2+3+4 is [(5)(5) - 5]/2, which easily generalizes to 1+2+3+...+n =
[(n+1)(n+1) - (n+1)]/2.
:[font = input; Cclosed; preserveAspect; startGroup]
Clear[s1,s2,s3,s4]
:[font = input; preserveAspect]
s1=Graphics[
{Line[{{0,0},{5,0}}],Line[{{0,1},{4,1}}],
Line[{{0,2},{3,2}}],Line[{{0,3},{2,3}}],
Line[{{0,4},{1,4}}],
Line[{{0,0},{0,5}}],Line[{{1,0},{1,4}}],
Line[{{2,0},{2,3}}],Line[{{3,0},{3,2}}],
Line[{{4,0},{4,1}}]}];
:[font = input; preserveAspect]
s2=Graphics[{GrayLevel[.25],
Rectangle[{0,4},{1,5}],
Rectangle[{1,3},{2,4}],
Rectangle[{2,2},{3,3}],
Rectangle[{3,1},{4,2}],
Rectangle[{4,0},{5,1}]}];
:[font = input; preserveAspect]
s3=Graphics[
{Line[{{0,5},{5,5}}],Line[{{0,4},{5,4}}],
Line[{{2,3},{5,3}}],Line[{{3,2},{5,2}}],
Line[{{4,1},{5,1}}],
Line[{{1,4},{1,5}}],Line[{{2,3},{2,5}}],
Line[{{3,2},{3,5}}],Line[{{4,0},{4,5}}],
Line[{{5,0},{5,5}}]}];
:[font = input; preserveAspect]
s4=Graphics[Text[FontForm[
"IntegerSum",{"Chicago",10}],
{2.5,5.3}]];
:[font = input; preserveAspect; endGroup]
Show[s1,s2,s3,s4,AspectRatio->Automatic];
:[font = text; inactive; Cclosed; pageBreak; noPageBreakInGroup; preserveAspect; startGroup]
Notice that Mathematica can easily convert
[(n+1)(n+1) - (n+1)]/2 to more familiar forms:
;[s]
3:0,0;12,1;23,0;91,-1;
2:2,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,2,12,0,0,0;
:[font = input; Cclosed; preserveAspect; startGroup]
Simplify[((n+1)(n+1)-(n+1))/2]
:[font = output; output; inactive; preserveAspect; endGroup]
(n + n^2)/2
;[o]
2
n + n
------
2
:[font = text; inactive; Cclosed; noPageBreakInGroup; preserveAspect; startGroup]
or
:[font = input; Cclosed; preserveAspect; startGroup]
Factor[%]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup; endGroup]
(n*(1 + n))/2
;[o]
n (1 + n)
---------
2
:[font = subsection; inactive; dontNoPageBreakBelow; noPageBreakInGroup; preserveAspect]
End Small Digression
:[font = section; inactive; Cclosed; noPageBreak; noPageBreakInGroup; preserveAspect; startGroup]
Finding the Sum 1Û+2Û+3Û+...+nÛ, the Legend Continues
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
Anyway, the entire "BigCube" is made up of three light-gray basic sum "pyramids," three "1+2+3+4" medium-gray separating sheets, and one "1+1+1+1+1 = 5" dark-gray main diagonal. Therefore 1Û+2Û+3Û+4Û is ’£Ú of [5Ü - 3(’Ý’Ó’ªÞ’Õ) - 5]. Again, this easily generalizes to 1Û+2Û+3Û+...+nÛ =
’£Ú[(n+1)Ü - 3(’ö’Ó’ö’ª±’Ú’Õ) - (n+1)]. Let's see if Mathematica can make this formula look more familiar:
;[s]
3:0,0;344,1;355,0;398,-1;
2:2,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,2,12,0,0,0;
:[font = input; Cclosed; preserveAspect; startGroup]
Simplify[((n+1)^3-3(n(n+1)/2)-(n+1))/3]
:[font = output; output; inactive; preserveAspect]
n/6 + n^2/2 + n^3/3
;[o]
2 3
n n n
- + -- + --
6 2 3
:[font = input; Cclosed; preserveAspect; startGroup]
Together[%]
:[font = output; output; inactive; preserveAspect]
(n + 3*n^2 + 2*n^3)/6
;[o]
2 3
n + 3 n + 2 n
---------------
6
:[font = input; preserveAspect; startGroup]
Factor[%]
:[font = output; output; inactive; preserveAspect]
(n*(1 + n)*(1 + 2*n))/6
;[o]
n (1 + n) (1 + 2 n)
-------------------
6
:[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup; endGroup]
There you go; the old familiar 1Û+2Û+3Û+...+nÛ = ’ö’Ó’ö’±’Ú’¤Õ’Ó’Û’ö’±’Ú’Õ.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Q.E.D.
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
[Which is an abbreviation for the semi-irrelevant (but pleasingly pretentious) Latin phrase "Quod erat demonstrandum," or "That which was to be demonstrated."]
:[font = text; inactive; preserveAspect; endGroup; endGroup]
[I did not make that up.]
^*)