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Stellar Numbers – Type 1

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Page 1: Stellar Numbers

Stellar Numbers – Type 1

Name: Nguyen

Page 2: Stellar Numbers

Introduction

The purpose of this task is to consider geometric shapes which lead to special numbers. Examples of special numbers are square numbers, 1, 4, 9, 16, which can be represented by squares of 1, 2, 3, and 4. The two geometric shapes that are examined in this portfolio are triangles and stars. The special numbers that are examined are triangular numbers and stellar numbers. General statements are produced from patterns these shapes generate. These statements are analyzed for their limitations and scope.

The portfolio is divided into two sections: “Triangular Numbers” and “Stellar Numbers”. The triangular shapes were created using Microsoft Paint and the stellar shapes were created using a Java-coded script (http://luckytoilet.wordpress.com/) created using Netbeans (http://netbeans.org/). Calculations are made using the TI-84 Plus graphic display calculator.

Triangular Numbers

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 …).

o Fig. 1

1 3 6 10 15

Complete the triangular numbers sequence with three more terms.

o Fig. 2

21 28 36

Find a general statement that represents the nth triangular number in terms of n.

In order to find the general statement, a table of values must be set up in the TI-84 Plus graphing display calculator. L1 represents “n” and L2 represents “y”.

o Fig. 3STAT > EDIT> Edit…

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Page 3: Stellar Numbers

The general statement will be generated using the calculator.

o Fig. 4STAT > CALC

The following table displays the various general statements generated by the calculator. To determine the general statement of the triangular pattern, n = 1, 2, 3, 4, 5, 6, 7, 8 is substituted into the statements in order to determine its validity. The most valid general statement therefore represents the nth triangular number in terms of n. (Note: Let x = n).

o Table 1General Statements Testing its Validity

y = 5n – 7.5

n y1 -2.52 2.53 7.54 12.55 17.56 22.57 27.58 32.5

y = 0.5n2 + 0.5n

n y1 12 33 64 105 156 217 28

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Page 4: Stellar Numbers

8 36

y = (1.069825054)(1.617929097)n

n y1 1.7309010842 2.8004752273 4.5309703554 7.3307887765 11.860696466 19.189765927 31.047680658 50.23294591

y = (0.940001486)(n)1.730969438

n y1 0.9400014862 3.1203424733 6.2952191054 10.358001875 15.241378766 20.897030317 27.287730648 34.38347028

From this information, the general statement for “QuadReg” is the most valid. The general statement of the triangular pattern is therefore:

The general statement is valid when n and when the shape is triangular. It does not apply for other ℝpolygons. For instance, the square numbers, 1, 4, 8, 16.

Stellar Numbers

Consider stellar (star) shapes with p vertices, leading to p-stellar numbers. The first four representations for a star with six vertices are shown in the four stages S1-S4 below. The 6-stellar number at each stage is the total number of dots in the diagram.

o Fig. 5

S1 S2 S3 S4

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Page 5: Stellar Numbers

Find the number of dots (i.e. the stellar number) in each stage up to S6. Organize the data so that you can reorganize and describe any patterns.

The information given above is organized in the table below. Let n = x = term number and Sn = y = number of dots.

o Table 2n Sn

1 1 1 = 1 + 12(0)2 13 13 = 1 + 12(1)3 37 37 = 1 + 12(3)4 73 73 = 1 + 12(6)5 121 121 = 1 + 12(10)6 181 181 = 1 + 12(15)

∴ Sn + 1 = Sn + 12(xn)12 is the number of vertices of S2. Xn values are equal to y-values in the “Triangular Pattern” task.

o Table 3n y1 12 33 64 105 15

Find an expression for the 6-stellar number at stage S7.

Since Sn+1 = Sn + 12(xn) and x=yThen S7 = S6+1 = S6 + 12(y6)

Find a general statement for the 6-stellar number at stage Sn in terms of n.

In order to find the general statement, a table of values must be set up in the TI-84 Plus graphing display calculator. L1 represents “n” and L2 represents “Sn”.

o Fig. 6STAT > EDIT> Edit…

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Page 6: Stellar Numbers

The general statement will be generated using the calculator.

o Fig. 7STAT > CALC

The following table displays the various general statements generated by the calculator. To determine the general statement of the 6-stellar pattern, n = 1, 2, 3, 4 is substituted into them to determine its validity. The most valid general statement therefore represents the 6-stellar number at stage Sn in terms of n. (Note: Let x = n and y = Sn).

o Table 4General Statements Testing its Validity

Sn = 24n-29

n Sn

1 -52 193 434 67

Sn = 6n2 - 6n + 1

n Sn

1 12 133 374 73

From this information, the general statement for “QuadReg” is the most accurate. The general statement for the 6-stellar number at stage Sn in terms of n is therefore:

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Page 7: Stellar Numbers

The general statement is valid when n and when the shape is stellar and when p = 6. It does not apply ℝfor other polygons.

Now repeat the steps above the other values of p.

The values that are used for p are 5, 7, and 8.

When p = 5

o Fig. 8

S1 S2 S3 S4

Find the number of dots (i.e. the stellar number) in each stage up to S6. Organize the data so that you can reorganize and describe any patterns.

The information given above is organized in the table below.

o Table 5n Sn

1 1 1 = 1 + 10(0)2 11 11 = 1 + 10(1)3 31 31 = 1 + 10(3)4 61 61 = 1 + 10(6)5 101 101 = 1 + 10(10)6 151 151 = 1 + 10(15)

∴ Sn + 1 = Sn + 10(xn)10 is the number of vertices of S2. Xn values are equal to y values in the “Triangular Pattern” task.

o Table 6n y1 12 33 64 105 15

Find an expression for the 5-stellar number at stage S7.

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Page 8: Stellar Numbers

Since Sn+1 = Sn + 10(xn) and x=yThen S7 = S6+1 = S6 + 10(y6)

Find a general statement for the 6-stellar number at stage Sn in terms of n.

In order to find the general statement, a table of values must be set up in the TI-84 Plus graphing display calculator. L1 represents “n” and L2 represents “Sn”.

o Fig. 9STAT > EDIT> Edit…

The general statement will be generated using the calculator.

o Fig. 10STAT > CALC

The following table displays the general statement produced by the calculator using “QuadReg”. To determine the general statement of the 5-stellar pattern, n = 1, 2, 3, 4 is substituted into it to determine its validity. (Note: Let x = n and y = Sn).

o Table 7General Statements Testing its Validity

Sn = 5n2 - 5n + 1

n Sn

1 12 113 314 61

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Page 9: Stellar Numbers

From this information, the general statement for “QuadReg” is valid. The general statement of the triangular pattern is therefore:

The general statement is valid when n and when the shape is stellar and when p = 5. It does not apply ℝfor other polygons.

When p = 7

o Fig. 11

S1 S2 S3 S4

Find the number of dots (i.e. the stellar number) in each stage up to S6. Organize the data so that you can reorganize and describe any patterns.

The information given above is organized in the table below.

o Table 8n Sn

1 1 1 = 1 + 14(0)2 15 15 = 1 + 14(1)3 43 43 = 1 + 14(3)4 85 85 = 1 + 14(6)5 141 141 = 1 + 14(10)6 211 211 = 1 + 14(15)

∴ Sn + 1 = Sn + 14(xn)14 is the number of vertices of S2. Xn values are equal to y values in the “Triangular Pattern” task.

o Table 9n y1 12 33 64 105 15

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Page 10: Stellar Numbers

Find an expression for the 7-stellar number at stage S7.

Since Sn+1 = Sn + 14(xn) and x=yThen S7 = S6+1 = S6 + 14(y6)

Find a general statement for the 7-stellar number at stage Sn in terms of n.

In order to find the general statement, a table of values must be set up in the TI-84 Plus graphing display calculator. L1 represents “n” and L2 represents “Sn”.

o Fig. 12STAT > EDIT> Edit…

The general statement will be generated using the calculator.

o Fig. 13STAT > CALC

The following table displays the general statement produced by the calculator using “QuadReg”. To determine the general statement of the 7-stellar pattern, n = 1, 2, 3, 4 is substituted into it to check its validity. (Note: Let x = n and y = Sn).

o Table 10General Statements Testing its Validity

Sn = 7n2 - 7n + 1

n Sn

1 12 153 434 85

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Page 11: Stellar Numbers

From this information, the general statement for “QuadReg” is valid. The general statement of the triangular pattern is therefore:

The general statement is valid when n and when the shape is stellar and when p = 7. It does not apply ℝfor other polygons.

When p = 8

o Fig. 14

S1 S2 S3 S4

Find the number of dots (i.e. the stellar number) in each stage up to S6. Organize the data so that you can reorganize and describe any patterns.

The information given above is organized in the table below.

o Table 11n Sn

1 1 1 = 1 + 16(0)2 17 17 = 1 + 16(1)3 49 49 = 1 + 16(3)4 97 97 = 1 + 16(6)5 161 161 = 1 + 16(10)6 241 241 = 1 + 16(15)

∴ Sn+1 = Sn + 16(xn)16 is the number of vertices of S2. Xn values are equal to y values in the “Triangular Pattern” task.

o Table 12n y1 12 33 64 105 15

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Page 12: Stellar Numbers

Find an expression for the 8-stellar number at stage S7.

Since Sn+1 = Sn + 16(xn) and x=yThen S7 = S6+1 = S6 + 16(y6)

Find a general statement for the 8-stellar number at stage Sn in terms of n.

In order to find the general statement, a table of values must be set up in the TI-84 Plus graphing display calculator. L1 represents “n” and L2 represents “Sn”.

o Fig. 15STAT > EDIT> Edit…

The general statement will be generated using the calculator.

o Fig. 16STAT > CALC

The following table displays the general statement produced by the calculator using “QuadReg”. To determine the general statement of the 8-stellar pattern, n = 1, 2, 3, 4 is substituted into it to check its validity. (Note: Let x = n and y = Sn).

o Table 13General Statements Testing its Validity

n Sn

1 12 173 494 97

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Page 13: Stellar Numbers

Sn = 8n2 - 8n + 1

From this information, the general statement for “QuadReg” is valid. The general statement of the triangular pattern is therefore:

The general statement is valid when n and when the shape is stellar and when p = 8. It does not apply ℝfor other polygons.

Hence, produce the general statement, in terms of p and n, that generates the sequence of p-stellar number for any value of p at stage Sn.

The following table organizes the information obtained regarding the 5, 6, 7, and 8 stellar number patterns. As shown in the previous areas of this investigation, the each of the general statements have proven to be valid.

o Table 14Stellar Number General Statement 5678

Based on these general statements, the general statement (in terms of p and n) that generates the sequence of p-stellar number for any value of p at stage Sn is:

Where “p” represents the stellar number and “n” represents the term number.

The general statement is valid when p and when the shape is stellar. It does not apply for other ℝpolygons.

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