Lacsap’s Fraction free essay sample

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. 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) Continued I decide to let numerator as the y-axis and nth row as the x-axis. We will write a custom essay sample on Lacsap’s Fraction or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Numerator Row number Fig. 2-1 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. Fig. 2-2 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. * 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). And this model really plays an important role in this task and provides me a more convenient method to find out the general statement. Within this task, in part B * Variables: the number of row and element number * Constraint:value of denominator Moreover, in the part B the mathematical model of integration by substitution plays a significant role, the general statement is getting through it.