Lacsaps 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 inthis fraction.How to find numeratorIn this project, the relationship between the row number, n, the numerator, and thedenominator of the pattern shown below. 1 1 1 1 1 1 1 1 1 1Figure 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 ofPascal’ 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.
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 15Caption: 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]
How to find denominator 1 )+0 1 )+0Figure 3: The pattern showing the difference of denominator and numerator for eachfraction. The first element and the last element are cut off since it is known that all ofthem are to be 1. However, only first row is not cut off.Table 1: The table showing the relationship between row number and difference ofnumerator and denominator for each 2nd element Row Number (n) Difference of Numerator and Denominator 1 0 2 1 3 2 4 3 5 4The difference of numerator and denominator increases by one. Moreover, it is clearthat the difference between row number and difference of numerator and denominator is1. Thus, the difference can be stated as (n-1). Therefore, the denominator of the 1stelement can be shown as, [Eq. 4]
Table 2: The table showing the relationship between row number and difference ofnumerator and denominator for each 1st element Row Number Difference of Numerator and Denominator 1 N/A 2 0 3 2 4 4 5 6The difference of numerator and denominator in 1st row is not applicable since there isno 2nd element in the 1st row. The difference of numerator and denominator increases bytwo. Thus, the difference can be stated as 2(n-2). Thus, the relationship betweendenominator 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 thoseequation 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
Thus, the denominator in 6th row can be solved as, - 1st element - 2nd element 13 - 3rd element 12 - 4th element 13 - 5th elementTherefore, the pattern in 6th row is
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,
ConclusionTherefore, 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 bothside of the given pattern should be cut out when the numerator is calculated. Thus, thesecond element of each row is counted as “the first element.” Second, n in generalstatement of numerator must be greater than 0. Third, the very first row of the givenpattern is counted as the 1st row.