- Level: International Baccalaureate
- Subject: Maths
- Word count: 1314
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.
Objectives
Our objectives are as follows
- Find the numerator in the sixth row.
- Find and plot the relation between the row number and the numerator in each row, write a general formula
- Find the 6th and 7th rows
Using the patterns in task 3, find the general statement for
. Where
- is the (r + 1) th element in the nth row, starting with r = 0.
- Find additional rows and test if the general statement agrees and is correct
- 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.
Middle
. I tried to find a relationship again using row 2 and found
for row 2.
Therefore,
and so
This can be rewritten as,
I used a graphic calculator to validate the general formula using a GDC and the QuadReg function.
Rewritten, or
Validating the formula manually
We can validate the formula by trying it on the numbers that we already know.
Finding the 6th and 7th rows
We can use the general formula to find the numerators of the 6th and 7th row.
6th row:
7th row:
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.
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.
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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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