IB Math SL Portfolio Stellar Numbers

8. Substitute with .

           

General statement of the  Triangular number

Stellar number images upto stage:

Stellar number images upto stage

I now put the above diagrams in numerical form:

I will now attempt to use quadratic regression to find out the general statement for the 6 stellar number at stage:

Calculator used: Texas Instruments-84 Silver Edition

1) Click

2. Select  and press  
3. I will now put in the values of  in List 1 and values of  in List 2.

4. Then click  again and select  
5. Select  and press
6. Press  and then  selecting   and then.
7. On pressing  We will see the following screen: 

8. Substitute  with  and  with

General Statement:

Expression for the 6 stellar number at stage, from the equation above:

 

 

At stage:  

Now the same for 5 and 7 Stellar shapes.

5 Stellar Shapes

Numerical form:

Calculator used: Texas Instruments-84 Silver Edition

1. Click
2. Select  and press  
3. I will now put in the values of  in List 1 and values of  in List 2.

       

4. Then click  again and select  
5. Select  and press
6. Press  and then  selecting   and then.
7. On pressing  We will see the following screen:  

8) Substitute  with  and  with

7 Stellar Shapes

Numerical form:

Calculator used: Texas Instruments-84 Silver Edition

1. Click
2. Select  and press  
3. I will now put in the values of  in List 1 and values of  in List 2.

 

4. Then click  again and select  
5. Select  and press
6. Press  and then  selecting   and then.
7. On pressing  We will see the following screen:   
8. Substitute  with  and  with

General Statement at 5 Stellar shapes:

General Statement at 6 Stellar shapes:

General Statement at 7 Stellar shapes:

 General Statement of p Stellar shapes:

I will now test this formula:

5 Stellar

(Previously Found.)

6 Stellar

(Previously Found)

7 Stellar

(Previously Found)

Conclusion

This formula can be applied to all p-stellar shapes only if they are positive natural numbers. However they can’t be applied to other polygons. There can’t be a stellar number for 1, 2, 3 or 4 as they do not have enough points form a star required. 4 HAS enough points but it would form a square and 3 forms a triangle.

The general statement is derived from the expressions formed in the cases of a 5, 6 and 7 stellar numbers. From there I understood that the numerical parts in each equation can be substituted with the stellar number itself hence finding the general statement, in terms of p and n that generates the sequence of p-stellar numbers for any value of p at stage Sn.

Bibliography

TI-84 Quadratic Regression method learnt from:

 

[Accessed on 21 February 2011]

Technology used:

Texas Instruments 84 Silver Edition calculator for Quadratic Regression

Microsoft Office 2010 & Microsoft Paint (Version 6.1) for shapes and figures

MathType (Version 6.7)