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LAcsap fractions - it is clear that in order to obtain a general statement for the pattern, two different statements will be needed

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Introduction

Jonghyun Choe

March 25 2011

Math IB SL

Internal Assessment – LASCAP’S Fraction

The goal of this task is to consider a set of fractions which are presented in a symmetrical, recurring sequence, and to find a general statement for the pattern.

The presented pattern is:

Row 1

                                                                                        1          1Row 2

                                                                                    1       32       1Row 3

                                                                               1       64       64       1Row 4

                                                                          1      107      106      107      1Row 5

                                                                     1      1511      159      159      1511     1

Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the values involved in the pattern. r represents the element number, starting at r=0, and n represents the row number starting at n=1. So for instance,   E52=159, the second element on the fifth row. Additionally,  N will represent the value of the numerator and D value of the denominator.

        To begin with, it is clear that in order to obtain a general statement for the pattern, two different statements will be needed to combine to form one final statement. This means that there will be two different statements, one that illustrates the numerators and another the denominators, which will be come together to find the general statement. To start the initial pattern, the pattern is split into two different patterns; one demonstrating the numerators and another denominators.

Step 2:

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Middle

=21

7th row numerator:                                 N(7)=0.5×72+0.5×7

N(7)=0.5×49+3.5

N(7)=28

Consequently, these are the values of numerators up to the 7th row.

1        1

3      3      3

6     6     6     6

10  10  10  10  10

15  15  15  15  15  15

21  21  21  21  21  21  21

28  28  28  28  28  28  28  28

Using the method in step 3 and equation 1 in figure 1, it is evident that the numerator in the 6th row is 21. Since both equations have brought same values, it can be concluded that equation 1 is a valid statement that demonstrates the pattern of the numerator. Equation 1 will be used later also, in order to form a general statement of the pattern of whole LACSAP Fractions.

Step 4: When examining the denominators in the LASCAP’S Fractions, their values are the highest in the beginning, decreases, and then increases again. For example, the denominators in row 5 are;  15  11  9  9  11  15.  From this pattern, we can easily see that the equation for finding the denominator would be in a parabola form.

Element

0

1

2

3

4

5

Denominator

15

11

9

9

11

15

The relationship between the denominator and the element number is graphically plotted and a quadratic fit determined, using loggerpro.

image01.png

Figure 2: This parabola describes the relationship between the denominator and element number.

The equation for the fit is : D = r2  - nr+r0 . In this equation, r refers to the element number starting from 0, and r0

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Conclusion

For example, E7 (3) = 2816=  0.5n2+0.5n r2  - nr+r0  = 0.5 × (7)2+0.5 × (7) 32  - 7×3+28  = 2816 . Here, it is clear that the formula is applicable.

In order to make sure that the general statement is valid, finding the additional rows of the recurring sequence of fractions by using the general statement above would be useful. Here, I chose to settle on 2 additional rows which are the 8th and 9th rows in the pattern.

8th row numerator:                                      N(8)=0.5×82+0.5×8

N(8)=0.5×64+4

N8=36

9th row numerator:                                      N(9)=0.5×92+0.5×9

N(9)=0.5×81+4.5

N9=45

8th row second and eighth denominator:  D = 12  - 8 ×1+36

                                                                          D = 1- 8+36

                                                                          D = 29

8th row third and seventh denominator:  D = 22  - 8 ×2+36

                                                                         D = 4- 16+36

                                                                         D = 24

8th row fourth and sixth denominator:  D = 32  - 8 ×3+36

                                                                      D = 9- 24+36

                                                                      D = 21

8th row fifth denominator:  D = 42  - 8 ×4+36

                                                  D   = 16- 24+36

                                                  D = 28

9th row second and ninth denominator:  D = 12  - 9 ×1+45

                                                                         D = 1- 9+45

                                                                         D = 37

9th row third and eighth denominator:  D = 22  - 9 ×2+45

                                                                       D = 4- 18+45

                                                                        D = 31

9th row fourth and seventh denominator:  D = 32  - 9 ×3+45

                                                                            D = 9- 27+45

                                                                            D = 27

9th row fifth and sixth denominator:       D = 42  - 9 ×4+45

                                                                       D = 16- 36+45

                                                                        D = 25

Thus, these are the fractions up to the 9th row.

1        1

1      32     1

1      6464     1

1      107 106 107   1

1        1511   159   159    1511    1

1     2116    2113    2112     2113     2116    1

1      2822    2818    2816    2816    2818     28221

                                     1       3629    3624    3621    3628     3621     3624     3629     1

                                  1       4537    4531    4527    4525     4525     4527     4531    4537     1

This shows that the general statement for the symmetrical, recurring sequence of fractions is valid and will continue to work.

...read more.

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