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Stellar Numbers math portfolio

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Introduction

Title Page, Page Numbers on bottom. This task concerns geometric shapes which lead to stellar, and other special numbers. The special numbers are the total number of dots in a diagram. For example, in a sequence: square numbers with tn values of 1, 4, 9, and 16 correspond to the term numbers 1, 2, 3 and 4, where n is an element of all positive integers. Such is the case with other special numbers, which may produce triangles, or other two dimensional shapes. The main types of special numbers considered in this task are stellar ones. Necessary? > I bet IB doesn’t care. All stars in this portfolio were created with a program titled Stellar-jar. u can talk about all the steps that are part of this portfolio  kinda like here’s what im going to do then do it and for the conclusion tell them what you did. Hence the last step to explain how you arrived at the general statement

 As an example, here are the first four terms of the square number sequence:

          t1                t2                           t3                                       t4

Triangular numbers are the first type of special numbers considered in this task. The term numbers 1, 2, 3, 4, 5 all correspond to the triangular numbers 1, 3, 6, 10 and 15. Here is the visual representation of this.

After the first 5 terms given by the task, by counting the number of dots in terms 6, 7, and 8, the values are 21, 28, and 36 respectively.

The general term could be found in two ways. The first is algebraically. Consider this sequence a relation where the independent variable is term number and the dependent is the number of dots. By calculating the second differences for this set of data (independent as term numbers n and dependent as number of dots, tn)

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Middle

a, b and c values will be given by the calculator.  

Now the task asks that the above steps are repeated for other values of p, shown below. It must be noted that in order for a star to be formed, creating stellar numbers, the sequence must have a p value of at least 3, because at positive integer values of 1 and 2, a star is not formed. Therefore is an element or positive integers.

First p=5 is considered. Below are stages 1-6

  S1                             S2                                 S3                                     S4

                       S5                                        S6

Now here is a chart showing the number of dots (the stellar number) in these stages, when p=5. Because these stellar numbers can be considered a sequence (where n is the term number, or the stage and Snis the number of dots or the stellar number, and n is an element of whole numbers) or a relation, it also shows up to the third differences.

n

Sn

1st differences

2nd differences

3rd differences

1

1

/

/

/

2

11

10

/

/

3

31

20

10

/

4

61

30

10

0

5

101

40

10

0

6

151

50

10

0

When the number of dots was counted, it was noticed that for each added layer there were ten more dots in that layer. Then after organizing the data into the above table, the value of the second differences (the degree of the relation created by these stellar numbers) was also 10. 10 is two times the number of vertices in this case.  Also it is noticed that once the second differences do exist (starting from when n= 3), at that n value, adding the Sn   value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.

An expression for the 7th term is S7= S6+ (S6-S5) + 10 (, where 10 is the constant second difference).

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Conclusion

n

Sn

1st differences

2nd differences

3rd differences

1

1

/

/

/

2

5

4

/

/

3

13

8

4

/

4

25

12

4

0

5

41

16

4

0

6

61

20

4

0

(The 2nd differences are 2 times the p value; the recursion formula still applies, as well as other patterns are the same).

 Yet, visually, only rhombii are formed, and therefore p=2 is inadmissible. Depends on how you define a stellar number.

To arrive at the general statement, there had to have been many trials done to get the specific general statementfor different p-values to notice that in the general equation for each of those values,theaandb values were the p value and the negative of the p value respectively. The c value was always one, no matter what the equation. For example, in Sn, this for for p=13, the c value is one, and it is evident that the a and b values are the p value and the negative of the p value respectively

Overall this task helps to further ones understanding of special numbers and their properties. From the ones considered in this task, it seems that the equations for all special numbers that create two-dimensional figures, the equations are to the second degree. More investigation should be done to prove this.

Header: Type 1, Stellar numbers, candidate #, mathematics SL > this is what IB survival told me

Footer: page ___ of ___ > again what IB survival told me

TITLE PAGE! (name candidate number type 1 stellar numbers school name) that should be enough

I recommend italicizing your variables and learn to use the equation editor. (Alt =). Also, be sure to format this better, state restrictions on your final equations (so post calculations e.g. if you’re saying THIS IS THE EQUATION FOR THIS). Also, for testing validity, use numbers in equations to test backed up with reasoning.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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