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Type I Internal Assessment (Lascap's Triangle)

Extracts from this document...

Introduction


Mathematics Coursework

Introduction

In this coursework, we have to look for a pattern in the numbers given to us in a triangular format as seen in the image below.

The image is a Lacsap’s Fraction. Lacsap’s fractions look a lot like Pascal’s Triangle and the work “lacsap” is “pascal” backwards which points further to their relationship.

image10.png

Objectives

Our objectives are as follows

  1. Find the numerator in the sixth row.
  2. Find and plot the relation between the row number and the numerator in each row, write a general formula
  3. Find the 6th and 7th rows

Using the patterns in task 3, find the general statement forimage00.png

. Where image00.png

  1.  is the (r + 1) th element in the nth row, starting with r = 0.
  2. Find additional rows and test if the general statement agrees and is correct
  3. Discuss the limitations of the general statement

Finding the Numerator of the sixth row

By looking at the Lacsap’s fraction and comparing it to the Pascal’s triangle, I observed that the numerators of the Lacsap’s fraction are the same as the third element number of the Pascal’s Triangle as seen in the image below. By this theory, the sixth numerator should be 21.image11.png

...read more.

Middle

. I tried to find a relationship again using row 2 and found image03.png

  for row 2.

Therefore, image04.png

 and so image05.png

This can be rewritten as, image06.png

I used a graphic calculator to validate the general formula using a GDC and the QuadReg function.image24.gifimage25.gif

Rewritten,                      or

Validating the formula manually

We can validate the formula by trying it on the numbers that we already know.

image26.gif

image27.gif

image28.gif

Finding the 6th and 7th rows

We can use the general formula to find the numerators of the 6th and 7th row.

6th row:

image29.gif

7th row:

image12.gif

Finding the denominator

After taking a hard look at the Lacsap’s Fractions, brain storming through all the possibilities and then through some trial and error, I found that the denominator too is related to the Pascal’s triangle. I have made an image which shows a Pascal’s triangle. In the image, I have highlighted the numbers which represent the difference in the numerator and the denominator of the Lacsap’s fractions.

image13.png

We also observe that the difference in the numerator and the denominator if we go diagonally for the first element number is 1, 2,3,4,5 respectively.

...read more.

Conclusion

Conclusion

I used mathematics and a few observations to find the different formulas required to find the final general formula. I have also used modern technology to validate the formula of my numerator. This was done using my GDC – QuadReg function to validate the formula before moving on to finding the denominator. Using some observation and trial and error, I was able to find a relationship between the numerator and the denominator which in turn allowed me to find a general formula for the denominator. Finally, after finding these two formulas, I combined them in a simple way to get the general formula and used this formula to find additional rows to prove the formula works.

...read more.

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