Lacsap's Fractions

Type I – Mathematical Investigation
Lacsap’s Fractions

The focus of this investigation is surrounding Lascap’s Fractions. They are a group of numbers set up in a certain pattern. A similar mathematical example to Lacsap’s Fractions is Pascal’s Triangle. Pascal’s Triangle represents the coefficients of the binomial expansion of quadratic equations. It is arranged in such a way that the number underneath the two numbers above it, is the sum.
Ex. 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

In the example of Pascal’s Triangle below, the highlighted numbers represent this pattern. The two numbers above the third add up to equal the third. (e.g. 2+1=3)

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

Although Pascal’s Triangle is set up similarly to Lacsap’s Fractions, the patterns are different and the numbers are set up in fractions as opposed to whole numbers. The first five rows of Lacsap’s Fractions are set up like this:

1

1 3/2 1

1 6/4 6/4 1

1 10/7 10/6 10/7 1

1 15/11 15/9 15/9 15/11 1

In the next part of the investigation, we will explore the patterns of Lacsap’s Fractions and how to find the following rows.
Part 1:

To begin the investigation, we are given the first five rows of Lacsap’s Fractions as shown below.

1 1

1 3/2 1

1 6/4 6/4 1

1 10/7 10/6 10/7 1

1 15/11 15/7 15/7 15/11 1

To figure out the various patterns, it would be easiest to split the fractions into two parts. Let’s start with including only the numerators in the same pyramid pattern. Notice that in each row, the numerator is constant throughout. Now, if