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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
-
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 Instruments-84 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 Instruments-84 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 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)