1882 Words8 Pages

Lascap’s Fraction

Arshbir Aulakh

Math IA

000277-0

International Baccalaureate

Sir Winston Churchill Secondary

Introduction

In this investigation, I will consider these sets of numbers that are present in a symmetrical pattern. To commence this task, we will have to consider the five rows of numbers shown below. 1 1

1 3/2 1

1 6/4 6/4 1

1 10/7 10/6 10/7 1

1 15/11 15/9 15/9 15/11 1

Figure 1: The given symmetrical pattern

These numbers are arranged in a Pascal’s triangle, which is a triangular array of binomial coefficients that was named after a French mathematician Blaise Pascal. Lascap’s Fraction is spelt backwards for Pascal’s fraction and there should be a way to solve this. There is an interesting pattern in figure one where in the 1st row, there are two numbers and in the 2nd row, 3 numbers in the 3rd row, and so forth. So through this, we know that there is a new number introduced in every consecutive row. But there is an easier way to represent this pattern and to solve for the Pascal’s triangle is by using this formula.

〖(x+y)〗^n=(n¦0) x^n y^0+(n¦1) x^n y^1+(n¦2) x^(n-2) y^2+⋯+(n¦(n-1)) x^1 y^(n-1)+(n¦n) x^0 y^n

If there happens to be a formula for Pascal’s fraction, then there should be a formula for tis Lascap’s Fraction. The first thing we will need to investigate is to describe how to find the numerator of the sixth row. This could be solved by adding the row number with the previous

Arshbir Aulakh

Math IA

000277-0

International Baccalaureate

Sir Winston Churchill Secondary

Introduction

In this investigation, I will consider these sets of numbers that are present in a symmetrical pattern. To commence this task, we will have to consider the five rows of numbers shown below. 1 1

1 3/2 1

1 6/4 6/4 1

1 10/7 10/6 10/7 1

1 15/11 15/9 15/9 15/11 1

Figure 1: The given symmetrical pattern

These numbers are arranged in a Pascal’s triangle, which is a triangular array of binomial coefficients that was named after a French mathematician Blaise Pascal. Lascap’s Fraction is spelt backwards for Pascal’s fraction and there should be a way to solve this. There is an interesting pattern in figure one where in the 1st row, there are two numbers and in the 2nd row, 3 numbers in the 3rd row, and so forth. So through this, we know that there is a new number introduced in every consecutive row. But there is an easier way to represent this pattern and to solve for the Pascal’s triangle is by using this formula.

〖(x+y)〗^n=(n¦0) x^n y^0+(n¦1) x^n y^1+(n¦2) x^(n-2) y^2+⋯+(n¦(n-1)) x^1 y^(n-1)+(n¦n) x^0 y^n

If there happens to be a formula for Pascal’s fraction, then there should be a formula for tis Lascap’s Fraction. The first thing we will need to investigate is to describe how to find the numerator of the sixth row. This could be solved by adding the row number with the previous

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