March 25 2011

Math IB SL

Internal Assessment – LASCAP’S Fraction

The goal of this task is to consider a set of fractions which are presented in a symmetrical, recurring sequence, and to find a general statement for the pattern.

The presented pattern is:

Row 1

1 1 Row 2

1 32 1 Row 3

1 64 64 1 Row 4

1 107 106 107 1 Row 5

1 1511 159 159 1511 1

Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the values involved in the pattern. r represents the element number, starting at r=0, and n represents the row number starting at n=1. So for instance, E52=159, the second element on the fifth row. Additionally, N will represent the value of the numerator and D value of the denominator.

To begin with, it is clear that in order to obtain a general statement for the pattern, two different statements will be needed to combine to form one final statement. This means that there will be two different statements, one that illustrates the numerators and another the denominators, which will be come together to find the general statement. To start the initial pattern, the pattern is split into two different patterns; one demonstrating the numerators and another denominators.

Step 2: This pattern demonstrates the pattern of the numerators. It is clear that all of the numerators in the nth row are equal. For example all numerators in row 3 are 6. 1 1

3 3 3

6 6 6 6

10 10 10 10 10

15 15 15 15 15 15

Row...

(1)