Lacsap
Introduction:
In this internal assessment, we will investigate a set of numbers that are presented in a symmetrical pattern. 5 rows of numbers are given in a shape of a triangle, therefore a connection can be made to Pascal’s triangle. Another hint can also easily be noticed as Lacsap is exactly the backwards of Pascal. The goal of the investigation is to find the general statement En(r), where En(r) is the (r+1)th element in the nth row, starting with r=0. An example of this would be . In order to develop the general statement for En(r), patterns have to be found for the calculation of the numerator and the denominator.
Figure 1: Lacsap’s fractions 1 1
1 3/2 1
1 6/4 6/4 1
1 10/7 10/6 10/7 1
1 15/11 15/9 15/9 15/11 1
Figure 2: Pascal’s triangle (n/r), where n represents the number of rows and r the number of the element
Calculation of the numerator:
Table 1: number of rows vs. numerator number of rows(n) numerator
1 1
2 3
3 6
4 10
5 15
Figure 3: number of rows vs. numerator The relationship between the number of rows and the numerator can be plot using a graph (Figure 3). The numerators of the first five rows are 1,3,6,10 and 15. The value of the numerator increases by one more each time, so the equation can be stated. a1 1 a2 3 a3 6 a4 10 a5 15
Here are some sample calculations based on the equation : n=2 n=5 Calculations of the numerators of the sixth and seventh rows: n=6 n=7 A pattern for the numerator can also be found using the two figures above (Figure 1,2). It can be noticed that the numerators of the fractions are always equal to the third elements in Pascal’s triangle, equal to the numbers that occur at r=2. Therefore the equation for calculating the numerator can be stated as , where n represents the number of rows. C2 corresponds to the second element in Pascal’s triangle.
Here are some sample calculations based on the statement: n=1
In this internal assessment, we will investigate a set of numbers that are presented in a symmetrical pattern. 5 rows of numbers are given in a shape of a triangle, therefore a connection can be made to Pascal’s triangle. Another hint can also easily be noticed as Lacsap is exactly the backwards of Pascal. The goal of the investigation is to find the general statement En(r), where En(r) is the (r+1)th element in the nth row, starting with r=0. An example of this would be . In order to develop the general statement for En(r), patterns have to be found for the calculation of the numerator and the denominator.
Figure 1: Lacsap’s fractions 1 1
1 3/2 1
1 6/4 6/4 1
1 10/7 10/6 10/7 1
1 15/11 15/9 15/9 15/11 1
Figure 2: Pascal’s triangle (n/r), where n represents the number of rows and r the number of the element
Calculation of the numerator:
Table 1: number of rows vs. numerator number of rows(n) numerator
1 1
2 3
3 6
4 10
5 15
Figure 3: number of rows vs. numerator The relationship between the number of rows and the numerator can be plot using a graph (Figure 3). The numerators of the first five rows are 1,3,6,10 and 15. The value of the numerator increases by one more each time, so the equation can be stated. a1 1 a2 3 a3 6 a4 10 a5 15
Here are some sample calculations based on the equation : n=2 n=5 Calculations of the numerators of the sixth and seventh rows: n=6 n=7 A pattern for the numerator can also be found using the two figures above (Figure 1,2). It can be noticed that the numerators of the fractions are always equal to the third elements in Pascal’s triangle, equal to the numbers that occur at r=2. Therefore the equation for calculating the numerator can be stated as , where n represents the number of rows. C2 corresponds to the second element in Pascal’s triangle.
Here are some sample calculations based on the statement: n=1