Triangle,Numbers,Pascal,One,th education Triangle Numbers in Pascals Triangle
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One of the most famous patterns that can be found in Pascal's triangle isthe triangle numbers, which can be found in the diagonals of Pascal's Triangle.In this article, I explain in a variety of different ways what triangle numbersare, as well as showing exactly where triangle numbers can be found in Pascal'striangle.Triangle numbers can be found by looking at the second diagonal in Pascal'striangle when it is drawn centrally. However, by left justifying the numbers inPascal's triangle, as in the diagram below, they can be found in the thirdcolumn: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 1The sequence in the third column begins 1,3,6,10,15,21. This is the sequencewere interested in. To help explain what this sequence is, I have shown twodifferent ways of thinking of it below:Method 1: The nth triangle number is the sum of all the positive wholenumbers up to n. This may sound confusing, but I will give a couple ofexamples. 21 is the seventh triangle number, because it is the sum of1+2+3+4+5+6+7. So what about the tenth number in the sequence? It is 1+2+3+4+5+6+7+8+9+10,which if you work it out is 55. Incidentally, you may have been given the taskat school before of adding up all of the numbers from 1 to 100 or 1 to 1000.What you are really being asked to do is find the 100th or 1000th trianglenumber (this can be done quickly in a variety of sneaky ways - unfortunatelyhowever, calculating all the rows of Pascal's triangle and looking down thethird column is NOT one of these methods!)Method 2: This method is much more exciting. It's a hands on, visual way ofunderstanding triangle numbers, and it also explains where this sequence getsits name from. You will need some coins or counters (or a pen and paper). Whatyou need to do is arrange your counters in triangles. You want to create anequilateral triangle (one with the same number of counters on each side), andyou need to "fill in" your triangle with counters, not just putcounters round the edges of the triangle. Now, if you were trying to find the6th triangle number, you would need to create a triangle with 6 counters oneach side. Once you have done that, you just have to count the number ofcounters you have used. You should find you get 21, which is correct (this canbe checked with method 1 or Pascal's triangle). You can try this out with anysized triangle of counters and it should always work. 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;}
Triangle,Numbers,Pascal,One,th