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Bionomial Investigation

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Introduction

Tran Quoc Hoang Viet

Math HL

EF International Academy 2009

Math Portfolio

  • Pascal Triangle is one of the most intersting way to place number in a logical pattern.This is done through very simple steps. Firstly, we start with the number 1 placed at the top.We then write down the following number on the lone below it,trying to form an imaginary triangle form. Each muber is calculated from the addition of the two numbers above it. The exceptions are the number “1” which are near the edge.
  • In this portfolio, I am going to find out and prove various formula and sums that are interconnected with each other through a common element: The Pascal Triangle. For certain time, I will have to prove certain symmetrical property, to find the sums, to work out the general formula and prove it is true, taking true examples of it. Explicating some of the formula connecting with the Pascal Triangle would be what I am about to do for most of my portfolio

The Pascal Triangle

                                                                                               1                                                                                                  Row 1

                                                                                        1           1                                                                                           Row 1

                                                                                1             2            1                                                                                   Row 2

                                                                         1                  3             3            1                                                                            Row 3

                                  1         4            6             4           1                                                                      Row 4        

                                   1           5            10         10           5             1                                     Row 5

                                                 1             6           15           20         15            6             1                                    Row 6

                                           1             7           21          35          35           21           7             1                            Row 7

                                    1          8             28         56           70          56           28           8             1                            Row 8

                            1            9            36          84         126         126          84          36            9             1                    Row 9

                     1           10          45          120       210         252         210         120        45          10            1                   Row 10

             1           11          55          165        330        462            462           330         165           55          11        1             Row 11

      1          12            66        220          495       792          924          792          495      220         66            12       1          Row 12

1          13         78          286        715       1287      1716       1716        1287        715          286        78          13       1 Row13

The binomial coefficients in the expansion of image00.pngimage00.png are defined as

image47.pngimage47.png

We have the equation of image12.pngimage12.png=image71.pngimage71.pngwhich is further derived from the above Triangle. I am now going to prove that this formula really work as true formula. Having prove this formula, will be able to get the subsequent number for a certain row

...read more.

Middle

image89.png

         A fraction within a fraction, the nominator can be moved to the upper nominator

          = image90.pngimage90.png + image89.pngimage89.png

(n-r+1)!Canbe further simplified through these steps:(n-r+1)! = (n-r+1)(n-r)!

                                                                                                                 (n-r)! = image91.pngimage91.png

          = image90.pngimage90.png + image92.pngimage92.png

A fraction within a fraction, the nominator can be moved to the upper nominator        

          = image90.pngimage90.png + image93.pngimage93.png

         Both fraction has the same denominator, they can join each other

          =  image94.pngimage94.png

          =  image95.pngimage95.png

          = image96.pngimage96.png

          = image97.pngimage97.png(as in the required form)

     Hence we can conclude thatimage98.pngimage98.png

Again, a general formula for the second sum, letter b, can be produced from the sum.That general formula isimage16.pngimage16.png + 2image16.pngimage16.png+image18.pngimage18.png. I am now going to use the proved formula and algebra rules to prove that the formula is the true formula for it.

 I split 2image20.pngimage20.png into 2 singles: image17.pngimage17.png

 = image02.pngimage02.png + image03.pngimage03.png +  image03.pngimage03.png + image04.pngimage04.png

Using the proved formula, image05.pngimage05.png+image06.pngimage06.png=image07.pngimage07.png

  = image08.pngimage08.png + image09.pngimage09.png

Using the proved formula, image10.pngimage10.png+image12.pngimage12.png=image13.pngimage13.png

  = image14.pngimage14.png

...read more.

Conclusion

image56.pngimage56.png. Thus, I am able to conduce to another general formula from the sums that I was given to do. These are the three formulas:

image10.pngimage10.png+image67.pngimage67.png + image68.pngimage68.png+image18.pngimage18.pngimage69.pngimage69.png + 3image16.pngimage16.png + 3image17.pngimage17.png +image18.pngimage18.png

and the sums:

image70.png

Once I got my hands on the answer for the sums, I then proceeded onto find the formula which connect any (k +1) successive coefficients in the nth row of the Pascal Triangle with the coefficient in the (n+k)th row. Should I am able to find it; I then, with the use of algebraic rule and mathematical induction, prove the formula is true. Besides using the general elements, I directly test the formulas through real number examples. Here are the two formulas that I managed to get:

image72.png

Consequently, I have to use real number sums to test these out. These are the sums that I did:

image73.png

The Pascal Triangle is one in a million mathematical wonders that human can ever think of. It connects all common senses of logic and algebra rules. From it, various general formulas can be derived from, making more mathematical properties at the same make them easier to understand. More formulas means that there will be less work of calculations required, making the job of a mathematician easier.  Although it is not just Blaise Pascal that got to think of such triangle, in the years of the 13th century, an arithmetic triangle which looks and works exactly in the same way as the Western Pascal triangle was produced by a Chinese mathematician .The only slight difference is that it was expressed through Chinese figures.

...read more.

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