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Lascap triangles Math Portfolio

Extracts from this document...

Introduction

Lascap’s fractions

Bill LI

IB Math SL

Internal Assessment Type 1

Table of Contents

Introduction

Pg.1

Finding the numerator

Pg.2

Graphical representation of correlation of the row number, n, and the numerator

Pg.3-4

Finding the sixth and seventh rows using patterns:

Pg.5

The general statement

Pg.6

Testing the validity of the general statement by finding additional rows

Pg.7

Conclusion

Pg.8

Introduction

Aim: In this task you will consider a set of numbers that are presented in a symmetrical pattern

Figure 1: The given pattern, Lascap’s fractions

1    1

1    image00.png

    1

1    image01.png

image01.png

    1

1    image02.png

image03.png

image02.png

    1

1    image04.png

image05.png

image05.png

image04.png

    1

Lascap’s fraction is in the form of a triangle where there is an increase of one element for each row that is increased. In this math portfolio Lascap’s fraction general formula will be disused and along with that how the general formula was formed due to patterns in the fraction themselves.

Finding the numerator:

First

...read more.

Middle

171

19

190

20

210

21

231

22

253

23

276

24

300

25

325

26

351

27

388

28

416

29

445

30

475

Graph 1:

The slope of the line in the graph best corresponds to a polynomial function. As the equation of the line is a sum notation.  

Finding the sixth and seventh rows using patterns:

Figure 3: Differences of each row’s denominators in Lascap’s fractions

image06.png

image06.png

image07.png

image00.png

image07.png

image08.png

image01.png

image01.png

image08.png

image09.png

image02.png

image03.png

image02.png

image09.png

image10.png

image04.png

image05.png

image05.png

image04.png

image10.png

The pattern found used to calculate the sixth and seventh rows is shown in figure 3. The numerator is calculated the same way as before with the equation, image11.png

 . The denominator is can also be calculated with a similar formula as the numerator. Going down diagonally to the right the number that is being added is increased by one each row. An example is in row 1,element 2 the denominator is increased by 1 as it goes to row 2,element 2. Then in row 2, element 2 the denominator is increased by 2 as it goes to row 3, element 2. This pattern keeps on going up to row 5, element 2.

...read more.

Conclusion

th element

There are however, limitations to the equation. The n and r cannot be a fraction, an imaginary number, or negative number. This is because n represents the nth row and r represents the (r+1)th element and neither the row number nor the element number can be a fraction unless the  fraction equals to one because the row number or the element can only be a whole number. The same applies to imaginary numbers or negative numbers. It is still possible to substitute a fraction or a negative number into the general statement and get an answer however it will not mean anything. Zero is only used to substitute for r in the general equation, but it can also be substituted for n. This substitution changes the general statement from image47.png

 to image60.png

 , which still allows the formula to operate the same way because essentially all it does is add a 0 to all the elements.

...read more.

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

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