
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 Instruments84 Silver Edition
1) Click

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

Then click again and select

Select and press

Press and then selecting and then.

On pressing We will see the following screen:

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 Instruments84 Silver Edition
 Click

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

Then click again and select

Select and press

Press and then selecting and then.

On pressing We will see the following screen:
8) Substitute with and with
7 Stellar Shapes
Numerical form:
Calculator used: Texas Instruments84 Silver Edition
 Click

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

Then click again and select

Select and press

Press and then selecting and then.

On pressing We will see the following screen:

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 pstellar 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 pstellar numbers for any value of p at stage Sn.
Bibliography
TI84 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)