# Math portfolio stellar numbers. This assessment will investigate geometric shapes that lead to special numbers.

Type 1

## Stellar numbers

Student: Alehsan petersen

Class: IB08A

This assessment will investigate geometric shapes that lead to special numbers.

An example of this would be square numbers (1,4,9,16…), which can be represented by squares of side (1,2,3,4…).

First, triangles will be investigated.

The number of dots (1,3,6,10,15…) in each triangle is a triangular number, which will be labelled Tn, the triangular sequence of n. One can see that these triangles are equilateral triangles, where the nth triangle has n number of dots on each side.

It can be observed that the increase in dots follows the pattern 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+…+n. Therefore one can derive other numbers for Tn.

Since one is dealing with a triangular shape it would be likely to investigate if the double amount of dots, a rectangular shape, gives a pattern. The sequence of the rectangle is called Rn.( 2,6,12,20,30……)

Observing Rn and the table one realizes that ever since d2  seems to be an arithmetic sequence, it is indicated that Tn itself could have a quadratic formula as its generator.Actually,one can  that the square of n plus n gives Rn. Expressing this mathematically,

Rn =

Since Rn=2Tn one can substitute Rn in order to get Tn,

2Tn=

2Tn=n(n+1)

Tn=

Thus it seems like a general statement that represents the nth triangular number in terms of n.

After investigating the triangular shapes, more complex Stellar shapes will be investigated.

And so forth…

Each star has a number of vertices these will be labelled p. Every value of p leads to P-Stellar number, labelled Sn. In this case it is a 6-stellar number.

The sequence Sn in this case is:

(1,13,37,73,121,181…)

The method  that was used for the table heading in cellA1 and B1.  And made an excel formula in A2 namely A1-B1. So the difference was obtained. For the graph A1 was plotted in relation to B1.

One can observe from the graph and the table that since difference of Sn values seem to be an arithmetic sequence then Sn is likely to be ...