# Lacsap's fraction math portfolio

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Introduction

Lacsap's fraction

Ryohei Kimura

IB Math SL 1

Internal Assessment Type 1

Lacsap’ fraction

Lacsap is backward word of Pascal. Thus, the Pascal’s triangle can be applied in this fraction.

How to find numerator

In this project, the relationship between the row number, n, the numerator, and the denominator of the pattern shown below.

1 | 1 | |||||||||||

1 | 1 | |||||||||||

1 | 1 | |||||||||||

1 | 1 | |||||||||||

1 | 1 |

Figure 1: The given symmetrical pattern

(Biwako)

Figure 2: The Pascal’s triangle shows the pattern of

.It is clear that the numerator of the pattern in Figure 1 is equal to the 3rd element of Pascal’ triangle which is when r = 2. Thus, the numerator in Figure 1 can be shown as,

(n+1)C2

[Eq.1]

where n represents row numbers.

Sample Calculation

- When n=1

(1+1)C2

(2)C2

-When n=2

(2+1)C2

(3)C2

-When n=5

(5+1)C2

(6)C2

15

Middle

x = 28 [Eq. 3]

How to find denominator

1 )+0 | 1 )+0 | ||||||||||||||

Figure 3: The pattern showing the difference of denominator and numerator for each fraction. The first element and the last element are cut off since it is known that all of them are to be 1. However, only first row is not cut off.

Table 1: The table showing the relationship between row number and difference of numerator and denominator for each 2nd element

Row Number (n) | Difference of Numerator and Denominator |

1 | 0 |

2 | 1 |

3 | 2 |

4 | 3 |

5 | 4 |

The difference of numerator and denominator increases by one. Moreover, it is clear that the difference between row number and difference of numerator and denominator is 1. Thus, the difference can be stated as (n-1). Therefore, the denominator of the 1st element can be shown as,

[Eq. 4]

Conclusion

1st element

2nd element

13

3rd element

12

4th element

13

5th element

Therefore, the pattern in 6th row is

Also, the denominator in 7th row can be solved as,

1st element

2nd element

18

3rd element

16

4th element

16

5th element

6th element

Hence, the pattern in 7th row is,

Conclusion

Therefore, the general statement of the rth element in nth row can be shows as,

[Eq.7]

where r is element number,

However, there are several limitations for this equation. First, number 1 located in both side of the given pattern should be cut out when the numerator is calculated. Thus, the second element of each row is counted as “the first element.” Second, n in general statement of numerator must be greater than 0. Third, the very first row of the given pattern is counted as the 1st row.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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