Sums,Rows,Pascal,Triangle,Pasc education Sums of Rows in Pascals Triangle
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Pascal's trianglecontains a vast range of patterns, including square, triangle and fibonaccinumbers, as well as many less well known sequences. In this article, however, Iexplain first what pattern can be seen by taking the sums of the row inPascal's triangle, and also why this pattern will always work whatever row itis tested for.Firstly, I have written out the first few rows of Pascal's Triangle andcalculated the sums of all the numbers in each row so that we can see if thereis a pattern:1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11+1=21+2+1=41+3+3+1=81+4+6+4+1=161+5+10+10+5+1=32You should begin to see a pattern emerging: The sums of the rows are thedoubling numbers 2,4,8,16,32, where each number is twice the previous one.Mathematically, we could write the sum of row n is 2^n (this means 2x2x2... ntimes. For example, 2^5 = 2x2x2x2x2, and 2^3 = 2x2x2.It's all very well spotting this intriguing pattern, but this alone is notentirely satisfactory for a mathematician. How do we know that this patternalways works? Checking it against more rows will not help because however manyrows we check, we cannot be sure it will work for the next one.We need a mathematical proof. It is this act of showing beyond any doubtsomething to be true or not true by a series of purely logical steps that setsmathematics apart from the other sciences. When we prove somethingmathematically, we can be absolutely certain it is always true, unlike apractical scientist who will carry out experiments (like our tests in the firstdiagram), and thus be only at most fairly certain of their results.We are going to prove (informally) this by a method called induction. Thismay sound scary, but in this case, its simple. Take any row on Pascal'striangle, say the 1, 4, 6, 4, 1 row. Now think about the row after it. We canwrite down the next row as an uncalculated sum, so instead of 1,5,10,10,5,1, wewrite 0+1, 1+4, 4+6, 6+4, 4+1, 1+0. You should be able to see that each numberfrom the 1, 4, 6, 4, 1 row has been used twice in the calculations for the nextrow. So we start with 1, 1 on row one, and each time every number is used twicein the following row, and hence the total of the rows of Pascal's trianglealways doubles. 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;}
Sums,Rows,Pascal,Triangle,Pasc