Pascals,Triangle,and,Cube,Numb education Pascals Triangle and Cube Numbers
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An amazingly wide range of sequences can be found in Pascal's triangle. Inaddition to triangle numbers, tetrahedral numbers, fibonacci numbers and squarenumbers, cube numbers are one sequence which is rarely discussed but can alsobe found in Pascal's triangle. In this article I explain where this sequencecan be found in Pascal's triangle, and why these seemingly unrelated ideas arelinked.To help explain where cube numbers can be found in Pascal's triangle, I willfirst briefly explain how the square numbers are formed. The third diagonal inof Pascal's triangle is 1,3,6,10,15,21... If we add together each of thesenumbers with its previous number, we get 0+1=1, 1+3=4, 3+6=9, 6+10=16... ,which are the square numbers. The way cube numbers can be formed from Pascal'striangle is similar, but a little more complex. Whilst the square numbers couldbe found in the third diagonal in, for the cube numbers, we must look at thefourth diagonal. The first few rows of Pascal's triangle are shown below, withthese numbers in bold:1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56 28 8 1This sequence is the tetrahedral numbers, whose differences give thetriangle numbers 1,3,6,10,15,21 (the sums of whole numbers e.g. 21 =1+2+3+4+5). However, if you try adding up consecutive pairs in the sequence1,4,10,20,35,56, you do not get the cube numbers. To see how to get thissequence, we will have to look at the formula for tetrahedral numbers, which is(n)(n+1)(n+2)/6. If you expand this, it you get (n^3 + 3n^2 + 2n)/6. Basically,we are trying to make n^3, so a good starting point is that here we have an^3/6 term, so we are likely to need to add together six tetrahedralnumbers to make n^3, not 2. Have a go at trying to find the cube numbers fromthis information. If you're still stuck, then look at the next paragraph.List the tetrahedral numbers with two zeros first: 0,0,1,4,10,20,35,56...Then, add three consecutive numbers at a time, but multiply the middle one by4:0 + 0 x 4 + 1 = 1 = 1^30 + 1 x 4 + 4 = 8 = 2^31 + 4 x 4 + 10 = 27 = 3^34 + 10 x 4 + 20 = 64 = 4^310 + 20 x 4 + 35 = 125 = 5^3This pattern does in fact, always continue. If you want to see why this isthe case, then try exanding and simplifying (n(n+1)(n+2))/6 + 4(n-1)(n)(n+1)/6+ ((n-2)(n-1)n)/6, which are the formulas for the nth, (n-1)th and (n-2)thtetrahedral numbers, and you should end up with n^3. Otherwise, as I expect isthe case (and I don't blame you), just enjoy the this interesting result andtest it out on your friends and family to find out if they can spot this hiddenlink between Pascal's triangle and cube numbers! Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;} Article Tags: Cube Numbers, Pascal's Triangle, Tetrahedral Numbers, Square Numbers
Pascals,Triangle,and,Cube,Numb