# Lacsap's Fraction Math Portfolio

by bluepoppyseed (student)

Lacsap’s Fraction

Thomas Ullrich

Internal Assessment Type 1

Math SL

Introduction:

In this IA Type 1 I’ll have to find patterns in a set of numbers, which are presented in a symmetrical pattern.

Step 1: Finding the numerator of the 6th row

At first, I had a look at the first 4 numerator, which are 3, 6, 10, 15. The first step between 6 and 3 is 3 and increases by 1 in each following row. Therefore we can say that the numerator in the 6th row is 21 (and the 7th is 28.)

Figure 1 Relation between the numerator and n

As you can see the numerator develops not linearly, but in some way exponentially.

Figure 2 Lacsap's Triangle

Step 2:

The next step is about finding the complete 6th and 7th row.  For the numerator, I found this specific pattern that describes it. As we see in our first graph, the numerator in row number 6 is 21 and in row number 7 it is 28.

Now it is about finding a pattern for the denominator.

To show the specific pattern for the denominator I marked the sequences which are responsible for the characteristical features of the Pascal’s triangle.

The sequences are:

2  (+2)  4 (+3)  7 (+4)  11

4  (+2)  6 (+3)  9

7  (+2)  9

Figure 4 Denominator pattern

As we can see in Figure 3, the denominator increases by 1 starting with a step of 2. Following this rule, the denominators are:

6th row:  16 - 13 - 12 - 13 - 16

7th row:  22 – 18 – 16 – 16 – 18 – 22

Since I have already found out the numerators (Step 1), the complete 6th and 7th row are as followed:

6th row :    1    1

7th row : 1    1

Step 3:

Step 3 is about finding a general expression for the numerator. In Figure 1, we see that the relation between the numerator and n is must be a nonlinear equation. So I use a general quadratic approach to find the exact expression for the numerator as a ...