This investigation is going to do research patterns relates to the Lacsap’s Fractions. For its external structure, Lacsap’s Fraction is analogous to Pascal’s Triangle. Lacsap’s Fraction presents the way of generating and organizing the binomial coefficients. Within this investigation, the work is planning to be divided into two parts. In the first part, the content will relate to the pattern of numerators. In the second part, I am going to do the research on the patterns of denominator and the general statement for. Admittedly, the technology of computing will be involved into this investigation (E.g. Geogebro and GSP5chs). The following figure 1-1 illustrates Lacsap’s Fraction.
* PART A - CALCULATIONS and ANALYSIS
Firstly, I am going to research the numerator patterns. By observing the numerators of these fractions, it is illustrated that the first row of numerator is 1, second row of numerator is 3, third row of numerator is 6, fourth row of numerator is 10 and fifth row of numerator is 15. Let’s present it into the mathematical way: (= numerator of the row)
I realize that , , , Thus it is easily to find the numerator of the sixth row which is getting from. In order to do further investigation, it is essential to make a data table. | numerator|
According to the table 2-1 and the foundational patterns present on the first page, it is sure there must be a mathematical relationship of row number and numerator number in between. To search the deeper relationship, the plot diagram is the meaningful and practical choice due to the shared feature with function diagram. I decide to let numerator as the y-axis and nth row as the x-axis. Numerator
By looking through the diagram (Fig.2-1), the shape of the curve seem to be similar with the quadratic function curve (See Fig. 2-2)
Secondly, let’s assume the relationship between the row number, n, and the numerator in each row is quadratic function. Next step, I am going to prove it.
Assuming their relationship is a quadratic function, and then chooses any 3 group of numbers to substitute into the equation. As x-axis represents the number of, y-axis represents the number of numerator. Chosen number group: (1,1), (2,3) and (3,6).
I am going to solve it by using Matrix method
, by using GC (graphic calculator) ()
Finally, substituting another group of data into f(x) ensure this function equation pattern expression of numerator is correct. Chosen number group: (6,21)
Thus, the pattern expression equation is exact correct and corresponds to the graph it illustrated. Let’s rewrite the equation to. According to the top equation we can directly substitute row number 7 into, the numerator which equals to.
* PART B - CALCULATIONS and ANALYSIS
In the part B, it will relate to pattern of the fraction’s denominator. And the pattern of denominator was presented by the general statement of , firstly let be the element in the row, starting with. Take an example first, according to the figure 3-1, the denominator of fourth row of the third number is 6, and we change it into the form which is. Fig.3-1
In the second step, looking at the denominators in each row, it is illustrated that the second row of denominator is 2, third row is 4, fourth row is 6 and 7, for fifth row of denominator is 15. In order to make it clear, I make a table (= denominator of the row). It is clear that the numbers of deno- | Denominator|
4| 7, 6, 7|
5| Table 3-1
11, 9, 9, 11|
minator does not remain the constant (table 3-1) . Therefore this time we are necessary to find some additional rows first (shown by the following figure 3-2, page 5).
Referring to figure 3-2, we add two more rows here...