- Level: International Baccalaureate
- Subject: Maths
- Word count: 1647
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 are defined as
We have the equation of =which 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
Middle
A fraction within a fraction, the nominator can be moved to the upper nominator
= +
(n-r+1)!Canbe further simplified through these steps:(n-r+1)! = (n-r+1)(n-r)!
(n-r)! =
= +
A fraction within a fraction, the nominator can be moved to the upper nominator
= +
Both fraction has the same denominator, they can join each other
=
=
=
= (as in the required form)
Hence we can conclude that
Again, a general formula for the second sum, letter b, can be produced from the sum.That general formula is + 2+. 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 2 into 2 singles:
= + + +
Using the proved formula, +=
= +
Using the proved formula, +=
=
Conclusion
+ + + + 3 + 3 +
and the sums:
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:
Consequently, I have to use real number sums to test these out. These are the sums that I did:
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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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