Image 1
Caption: Image 1 shows the set of data. L1 represents number of row (n) while L2 represents the value of numerator (N)
Image 2
Caption: Image 2 shows the range set which is -2≤ n ≤ 6 for the x-axis while -2 ≤ y ≤ 20 for the y-axis
Image 3
Caption: Image 3 shows the general equation obtained by using Quadratic Regression. Based on image 3, the general equation to find numerator is N = 0.5n2 + 0.5n.
Image 4
Caption: Image 4 shows the general equation to find the numerator on the Y-editor.
Image 5
Graph
y-axis : The value of numerator, N
x-axis : Rows, n
Caption: Image 5 shows the graph plotted using the general equation to find the numerator that was made earlier which is N= 0.5n2 + 0.5n
From the above graph, the equation to find the numerator shows a quadratic shaped graph which suggests the highest power of n = 2. The graph also shows that, as the number of n increases, the number of N gets larger. The irregular increment from one point to another makes the shap of the graph to be quadratic.
Thus, the general statement to find the numerator can be defined as:
N = 0.5n2 + 0.5n
= n2 + n
=
=
- Find the sixth and seventh rows.
1 1
1 1
1 1
1 1
1 1
Numerator
To calculate the numerator of the sixth and seventh row, the same way of calculating is used as described in the previous question. Like what have been described in the first question, the numerator of the sixth row is:
1 21 21 21 21 21 1
For the seventh row, the numerator can be calculated by adding 7 to the numerator of the sixth row; 21. The numerator of the seventh row is as follows:
1 28 28 28 28 28 28 1
Denominator
Referring back to the triangle given,
The denominator also forms a pattern similar to that of the numerator. The difference of the denominator between columns increases by 1.
Key :
-
n = Number of row
- D = Value of denominator
The table above shows the proof of the increment by 1 in the difference between denominators from different rows. Besides that, an attempt to predict the value of the denominator for the sixth and seventh row is also shown. Since the triangle is symmetrical, the value of the denominator for the sixth row is:
1 16 13 12 13 16 1
Following the symmetrical pattern from the previous rows of the triangle, the value of the denominator for the seventh row is:
1 22 18 16 16 18 22 1
Thus, the elements (E) of the sixth and seventh rows are:
n = 6 1 1
n = 7 1 1
-
Find the general statement for En(r).
Explanation on the general statement.
En(r) is the element from nth row and rth column. Every element in the given triangle is in a fraction form and thus, each element is consist of a numerator (N) and a denominator (D) in the form of for the respective row and column. Thus, the general equation of En(r) must consists of . The general equation to find the numerator is already discovered in the early part of this portfolio where the general equation is:
N =
Finding the general equation of the denominator.
The values of elements in the triangle all gives us clues on the patterns and correlations between one another. The difference of the numerator and denominator for elements in the same row also forms a pattern. Instead of finding the difference of denominators between rows, denominators can also be calculated by substracting the value of the difference between the numerator and denominator (N – D) from the value of numerator (N). The table belows gives a clearer picture of the claim made in the earlier sentence.
Equal values
Equal values
The table above proves that the value of denominator can be derived from subtracting the difference of the numerator and denominator (N – D) from the numerator (N);
D = N – (N – D)
Since the general equation of N is known, now, I only need to find any correlation between the row number, column number and (N – D). This is to make it easier to formulate the general equation of the denominator. After a few trials, the value of (N – D) is actually the result of multiplying the difference between n and r with r, r(n – r). Table below gives a clearer picture of the newly derived equation for (N – D).
Equal values
Equal values
The table above just proves that (N – D) can be simplified into r(n – r).
If D = N – [r(n – r)]
D = - [r(n – r)]
If general statement of En(r) =
En(r) = [ ]
- Finding additional rows using the general statement (test validity).
Using the general statement En(r) = [ , the elements for rows n = 8 and n = 9 are calculated to prove the that the general statement used is valid and correct.
For n = 8,
E8(0) = [ ]
=
E8(1) = [ ]
=
E8(2) = [ ]
=
E8(3) = [ ]
=
E8(4) = [ ]
=
E8(5) = [ ]
=
E8(6) = [ ]
=
E8(7) = [ ]
=
E8(8) = [ ]
=
For n = 9,
E9(0) = [ ]
=
E9(1) = [ ]
=
E9(2) = [ ]
=
E9(3) = [ ]
=
E9(4) = [ ]
=
E9(5) = [ ]
=
E9(6) = [ ]
=
E9(7) = [ ]
=
E9(8) = [ ]
=
E9(9) = [ ]
=
Arranging them in the form of the triangle,
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
When rows n = 8 and n = 9 are combined to the pre-made triangle, the triangular shape and the symmetrical order within the triangle are maintained. With the constant pattern being repeated, this proves that the general statement for En(r) which is [ is valid.
- Scope and/or limitations of the general statement.
-
The general statement is not applicable for row n = 0. Looking at the general statement, [, if value n = 0 is plugged into the equation, the solution will be 0 based on the rule that states that, any values that is multiplied by 0 equals to 0. Besides that, when the value of n = 0, the numerator will always be 0 and in the end will always give 0 as the answer. Thus, this general statement will not provide the element for n = 0, r = 0.
Example:
When n = 0, r = 1
E0(1) = [
=
= 0
- Since the pattern of the elements form a triangle which starts with column n = 1, the value of n < 0 will be undefined. Thus, plugging in negative values of n will no longer follow the original triangular shape.
-
The general stament is also is limited to column r0. Any negative values for r will no longer match the symmetrical pattern of the triangle.
CONCLUSION
The regular patterns shown throughout the triangle allows us to figure out the upcoming rows of the triangle. General equations of the numerator and the denominator are able to derived and thus, makes it easy to calculate the elements in other rows and columns. However, there are still limitations to this general statement. Since the shape that the symmetrical elements form is a triangle, the range of elements that can be calculated is limited to within the shape of the triangle only. For example, catering for n < 0 will no longer follows the triangle shape. All in all, observing patterns that the fractions potray enables us to discover what lies beneath all the numbers.