Lacsap

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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

11

13/21

16/46/41

110/710/610/71

115/1115/915/915/111

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
11
23
36
410
515

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. a11

a23
a36
a410
a515

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

n=2

n=4

Calculations of the numerator of the sixth and seventh rows:
n=6

n=7

So the numerator of the sixth row is 21, and the numerator for the seventh row is 28. Calculation of the denominator:
In order to find the general statement for the denominator, the relationship between the number of rows and the difference of numerator and denominator has to be examined. By looking at Lacsap’s fractions, the conclusion can be drawn that the first and last elements of each row but the first one can be eliminated, as the difference of the numerator and the denominator is always 1 for these elements. Table 2: number of rows vs. difference of numerator and denominator for each 1st element(r=1) number of rows(n)numeratordenominatordifference of numerator and denominator 1110

2321
3642
41073
515114

Patterns can be noticed by examining the table above. The difference between the numerator and the denominator increases by one for the elements that occur at r=1. Using this information, general statement of the denominator can be stated as Denominator1 , where n is the number of rows. Here are some sample calculations based on the general statement above:

n=2

n=3

n=5

Table 3: number of rows vs. the difference of numerator and denominator for each 2nd element(r=2) number of rows(n)numeratordenominatordifference of numerator and denominator 1---
2110
3642
41064
51596

A pattern can be noticed by examining the table above. The difference of the numerator and denominator of the elements that occur at r=2 increases by two this time. There is no value for the row number 1, as there is no 2nd element in row number 1. According to this, the general statement can be stated as Denominator2 , where n is the number of rows. Here are some sample calculations based on the statement above:

n=3

n=5

Table 4: number of rows vs. the difference of numerator and...
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