Fracciones Lacsap

Lacsap's fraction

Ryohei Kimura
IB Math SL 1
Internal Assessment Type 1

Lacsap’ fraction
Lacsap is backward word of Pascal. Thus, the Pascal’s triangle can be applied in this fraction.
Howto 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 | |32 | | 1 | | | |
| | 1 | | 64 | | 64 | | 1 | | |
| 1 | | 107 | 106 | 107 | 1 | |
1 | | 1511 | 159 | 159 | 1511 | 1 |
Figure 1: The given symmetrical pattern

(Biwako)
Figure2: The Pascal’s triangle shows the pattern ofnr

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 inFigure 1 can be shown as,

(n+1)C2 =numorator [Eq.1]
where n represents row numbers.
Sample Calculation
- When n=1
(1+1)C2 =x
(2)C2 =1
-When n=2
(2+1)C2 =x
(3)C2 =3
-When n=5(5+1)C2 =x
(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 =x
(7)C2 =x
x = 21[Eq. 2]
and the numerator of 7th row also can be found by,
(7+1)C2 =x
(8)C2 =x
x = 28 [Eq. 3]

How to find denominator
| 1 )+0 | 1 )+0 ||
| | | | | 32 )+1 | | | | |
| | | | 64 )+2 | 64 )+2 | | | |
| | | 107 )+3 | 106 )+4 | 107 )+3 | | |
| | 1511 )+4 | 159 )+6 | 159 )+6 | 1511 )+4 | |

Figure3: 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, onlyfirst 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...