This document presents the analysis of a pattern of fractions known as LASCAP's Fractions. [1] The pattern is split into the numerators and denominators which both follow quadratic relationships. [2] Equations are derived to express the numerators and denominators as functions of the row and element numbers. [3] Using these equations, additional rows are generated, verifying that the equations accurately predict the entire recurring sequence.

1. Jonghyun Choe
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
Row 2
Row 3
Row 4
Row 5
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, , 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.

2. 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 number (n) 1 2 3 4 5
Numerator (N) 1 3 6 10 15
N(n+1) - Nn N/A 2 3 4 5
Table 1: The increasing value of the numerators in relations to the row number.
From the table above, we can see that there is a downward pattern, in which the numerator
increases proportionally as the row number increases. It can be found that the value of N (n+1)
- Nn increases proportionally as the sequence continues.
The relationship between the row number and the numerator is graphically plotted and a
quadratic fit determined, using loggerpro.
Figure 1: The equation of the quadratic fit is the relationship between the numerator and
the row number. The equation for the fit is: N= or Equation 1

3. In this equation, N refers to the numerator. Therefore, N= or 0 is a
statement that represents step 2 and also step 1.
Step 3: In relation to table 1 and step 2, a pattern can be drawn. The difference between the
numerators of two consecutive rows is one more than the difference between the previous
numerators of two consecutive rows. This can be expressed in a formula N(n+1) - N(n) = N(n) -
N(n-1) + 1. For instance, N(3+1) - N(3) = N(3) - N(2) +1.
Using this method, numerator of 6th and 7th row can be determined. To find the 6th row’s
value, n should be plugged in as 5 so that N(6) can be found. As for the 7th row’s numerator, n
should be plugged in as 6.
6th row numerator is therefore: N(5+1) - N(5) = N(5) - N(4)+1
N(6) – 15 = 15 – 10+1
N(6) = 15+6
N(6) = 21
7th row numerator is therefore: N(6+1) - N(6) = N(6) - N(5)+1
N(7) – 21 = 21 – 15 +1
N(6) = 42 – 15 + 1
N(6) = 28
Not only by this method, but from the equation found in step 2, figure 1, 6th and 7th row
numerator can be found also.
6th row numerator:
7th row numerator:
.5

4. Consequently, these are the values of numerators up to the 7th row.
1 1
3 3 3
6 6 6 6
10 10 10 10 10
15 15 15 15 15 15
21 21 21 21 21 21 21
28 28 28 28 28 28 28 28
Using the method in step 3 and equation 1 in figure 1, it is evident that the numerator in the
6th row is 21. Since both equations have brought same values, it can be concluded that
equation 1 is a valid statement that demonstrates the pattern of the numerator. Equation 1
will be used later also, in order to form a general statement of the pattern of whole LACSAP
Fractions.
Step 4: When examining the denominators in the LASCAP’S Fractions, their values are the
highest in the beginning, decreases, and then increases again. For example, the
denominators in row 5 are; 15 11 9 9 11 15. From this pattern, we can easily see that the
equation for finding the denominator would be in a parabola form.
Element 0 1 2 3 4 5
Denominator 15 11 9 9 11 15
The relationship between the denominator and the element number is graphically
plotted and a quadratic fit determined, using loggerpro.

5. Figure 2: This parabola describes the relationship between the denominator and element
number.
The equation for the fit is : D = . In this equation, r refers to the element
number starting from 0, and being the first denominator value in the row. n refers to the
row number starting from 1.
To see if this equation work, the 3rd denominator value in the 3rd row was measured.
D=
D = 4 – 6 +6
D=4
With this equation, it is evident that the 6th and 7th row denominator values can be found.
We already know the first and last denominators from when numerators were found; which
are 21 and 28.
6th row second and sixth denominator: D =
D=
D = 16
6th row third and fifth denominator: D=
D=
D = 13

6. 6th row fourth denominator: D=
D=
D = 12
7th row second and seventh denominator: D=
D=
D = 22
7th row third and sixth denominator: D=
D=
D = 18
7th row fourth and fifth denominator: D=
D=
D = 16
Now, since the denominators in the 6th and 7th row are found, the sixth and seventh rows
can be drawn and added in the LACSAP’S Fractions.
Consequently, these are the fractions up to the 7th row.
1 1
1 1
1 1
1 1
1 1
1 1
1 1

7. Now that the patterns for the LASCAP’S Fractions are found, all fractions can be expressed in
the form when it is the (r+1)th element in the nth row, starting with r=0.
The general statement of the pattern is clearly found when using the equations for the
nominator and the denominator. Therefore, the general statement for will be
=
In order to see if the equation works correctly, we can plug in number and figure out if the
general statement works out.
For example, = = = = . Here, it is clear that
the formula is applicable.
In order to make sure that the general statement is valid, finding the additional rows of the
recurring sequence of fractions by using the general statement above would be useful. Here,
I chose to settle on 2 additional rows which are the 8th and 9th rows in the pattern.
8th row numerator:
9th row numerator:
8th row second and eighth denominator: D =
D=
D = 29
8th row third and seventh denominator: D =
D=
D = 24
8th row fourth and sixth denominator: D =
D=
D = 21
8th row fifth denominator: D =
D =
D = 28

8. 9th row second and ninth denominator: D =
D=
D = 37
9th row third and eighth denominator: D =
D=
D = 31
9th row fourth and seventh denominator: D =
D=
D = 27
9th row fifth and sixth denominator: D=
D=
D = 25
Thus, these are the fractions up to the 9th row.
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
This shows that the general statement for the symmetrical, recurring sequence of fractions
is valid and will continue to work.