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Lascap Fractions. In Lacsap's Fractions, when looking for a general pattern for the numerator, it can be noted that it does not increase linearly but exponentially

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Introduction

In Lacsap's Fractions, when looking for a general pattern for the numerator, it can be noted that it does not increase linearly but exponentially. Numerators are 3,6,10, and 15, each preceding numerator added by one plus the row number. Using this general statement it can be concluded that the numerator in the 6th row is 21 (15+6), and 28 for the 7th.

image06.png

Generating a Statement for the Numerator:

To generate an equation for the numerator of the fraction, the fraction data must be organized and graphed. The table below shows the relationship between the row number and numerator being relative to an exponential function as the sequence goes on. N(n+1)-Nn

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Middle

6th:

N(5+1)-N(5)=N(5)-N(4)+1

N(6)-15=15-10+1

N(6)=15+6

N(6)= 21

7th:

N(6-1)-N(6)=N(6)-N(5)+1

N(7)-21=21-15+1

N(7)=42-15+1

N(7)= 28

This is only a supplement to the equation found in the graph above (N=0.5n2+0.5n). This pattern only tests the validity of the equation derived from the table because of both methods concluding to the same value.

Generating a Statement for the Denominator:

To examine the denominators in Lascap's Fractions, the values for the 6th row and their corresponding elements were put onto a table, and ultimately a graph. Showing a pattern, it was concluded that the denominator could be found with a general equation of D=r2-nr+r0.

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Conclusion

: D=22-(5)(2)+15=11

6th: D=22-(6)(2)+21=13

After knowing how to determine both denominator and numerators, the Lacsap's Fraction triangle could be filled out to the 7th row. image09.png

General Statement Overall:

After determining equations for both numerator and denominator and following patterns, all Lacsap's Fractions could be found with a general statement of:

En(r)= image00.png

=image01.png

Although proven for each separate category of numerator and denominator, the general statement can be concluded valid with these tests.

Finding X row:

5th row 4th Fraction: E5(4)=image02.png

=image03.png

8th row 3rd Fraction: E8(4)=image04.png

=image05.png

Limits and Conditions:

Other than the Nth row having to be greater than 0, there is no limit to this equation as it can find any fraction within Lacsap's Fraction Triangle.

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

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