Ib Math Stellar Numbers Portfolio

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• Topic: Triangular number, Numbers, Polygonal number
• Pages : 6 (1232 words )
• Published : March 3, 2012

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Introduction: In this following assignment, I will be considering geometric shapes that lead to special numbers. The simplest examples of these are square numbers (1, 4, 9, 16, etc), which are derived from squaring 1, 2, 3, and 4. From this I got the equation y= x2. This equation is illustrated in the table below.

y=x2
|x |y |
|1 |1 |
|2 |4 |
|3 |9 |
|4 |16 |

In the table on the left, I observe that from the y value 1 to the y value 4 there is an increase of 3. From the y values 4 to 9, there is an increase of 5. From the y values 9 to 16, there is an increase of 7. This shows that it goes: +3, +5, +7, which is then increasing by 2 between each of those numbers.

Below, is the graph of y=x2

[pic]

The equation y=x2 comes from the general equation y= ax2-bx+c. Y=x2 is the same as y=x2+0x+0. Therefore, a=1, b=0, and c=0.

The next example I am going to show you is similar to the one above. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, 10, 15, 21, 28, 36).

|Tn |y | |1 |1 | |2 |3 | |3 |6 | |4 |10 | |5 |15 | |6 |21 | |7 |28 | |8 |36 |

In the table on the left, I observe that from the y value 1 to the y value 3 there is an increase of 2. From the y values 3 to 6, there is an increase of 3. From the y values 6 to 10, there is an increase of 4. This shows that it goes: +2, +3, +4, which is then increasing by 1 between each of those numbers.

This follows the form of an arithmetic sequence, because each new term is equal to the sum of the previous term plus the term number. A formula for this sequence can then be derived form the formula of an arithmetic sequence, shown below. n represents the term number, a represents the number of dots in the first term, and d represents the common difference between the terms.

[pic]
Since the first team has one dot, this means that for this sequence, a=1. The difference between each term is represented by d. Therefore, d=1. By substituting these values into the arithmetic sequence formula, I get the equation: [pic]

Which is then simplified to:
[pic]
This formula above can be used to calculate the number of dots of any triangular number is the sequence, n being greater than 0.

Graphing the points from the table above, gives me this graph:

The equation:
[pic]

is correct because the points in the table fall exactly on the graphed line.

Above, the equation: [pic] also follows the general equation y=ax2+bx+c. If I move around the equation to fit the general form, [pic]. Therefore, a= ½, b= ½, and c=0.

Next, I will be considering stellar shapes with p vertices, leading to p-stellar numbers. I will be showing you stages S1-S6 in diagrams below.

|Stage number |Number of dots |Dot increase between stages | |S1 |1 | | |S2 |13 |+12 | |S3 |37...