This document examines Lacsap's fraction, which is based on Pascal's triangle. It establishes relationships between the row number (n), element number (r), numerator, and denominator of each term. Specifically: (1) The numerator of the rth element in the nth row is given by (n+1)Cr. (2) The denominator of the rth element in the nth row is given by nr. (3) These relationships are demonstrated through sample calculations and analysis of patterns in the fractions. The document concludes by stating the general relationships between n, r, and the numerator and denominator in Lacsap's fraction.

1. Lacsap's fraction
Ryohei Kimura
IB Math SL 1
Internal Assessment Type 1

2. Lacsap’ fraction
Lacsap is backward word of Pascal. Thus, the Pascal’s triangle can be applied in
this fraction.
How to find numerator
In this project, the relationship between the row number, n, the numerator, and the
denominator of the pattern shown below.
1 1
1 1
1 1
1 1
1 1
Figure 1: The given symmetrical pattern
(Biwako)
Figure 2: The Pascal’s triangle shows the pattern of
.It is clear that the numerator of the pattern in Figure 1 is equal to the 3rd element of
Pascal’ triangle which is when r = 2. Thus, the numerator in Figure 1 can be shown as,
(n+1)C2 [Eq.1]
where n represents row numbers.

3. Sample Calculation
- When n=1
(1+1)C2
(2)C2
-When n=2
(2+1)C2
(3)C2
-When n=5
(5+1)C2
(6)C2 15
Caption: The row numbers above are randomly selected within a range of 0≤x≤5.
Therefore, the numerator of 6th row can be found by,
(6+1)C2
(7)C2
x = 21 [Eq. 2]
and the numerator of 7th row also can be found by,
(7+1)C2
(8)C2
x = 28 [Eq. 3]

4. How to find denominator
1 )+0 1 )+0
Figure 3: The pattern showing the difference of denominator and numerator for each
fraction. The first element and the last element are cut off since it is known that all of
them are to be 1. However, only first row is not cut off.
Table 1: The table showing the relationship between row number and difference of
numerator and denominator for each 2nd element
Row Number (n) Difference of Numerator
and Denominator
1 0
2 1
3 2
4 3
5 4
The difference of numerator and denominator increases by one. Moreover, it is clear
that the difference between row number and difference of numerator and denominator is
1. Thus, the difference can be stated as (n-1). Therefore, the denominator of the 1st
element can be shown as,
[Eq. 4]

5. Table 2: The table showing the relationship between row number and difference of
numerator and denominator for each 1st element
Row Number Difference of Numerator
and Denominator
1 N/A
2 0
3 2
4 4
5 6
The difference of numerator and denominator in 1st row is not applicable since there is
no 2nd element in the 1st row. The difference of numerator and denominator increases by
two. Thus, the difference can be stated as 2(n-2). Thus, the relationship between
denominator and numerator can be shown as,
[Eq.5]
where n is row number.
From these data, it can be detected that there is a pattern that the number used in those
equation is same as the element number. Therefore, the denominator can be stated as,
[Eq.6]
where n represents the row number and r represents the element number.
Sample Calculation
-When n = 4, and r = 3

6. Thus, the denominator in 6th row can be solved as,
- 1st element
- 2nd element
13
- 3rd element
12
- 4th element
13
- 5th element
Therefore, the pattern in 6th row is

7. Also, the denominator in 7th row can be solved as,
- 1st element
- 2nd element
18
- 3rd element
16
- 4th element
16
- 5th element
- 6th element
-
Hence, the pattern in 7th row is,

8. Conclusion
Therefore, the general statement of the rth element in nth row can be shows as,
[Eq.7]
where r is element number,
However, there are several limitations for this equation. First, number 1 located in both
side of the given pattern should be cut out when the numerator is calculated. Thus, the
second element of each row is counted as “the first element.” Second, n in general
statement of numerator must be greater than 0. Third, the very first row of the given
pattern is counted as the 1st row.