Stellar Numbers Portfolio. The simplest example of these is square numbers, but over the course of this investigation, both triangular numbers and stellar numbers will be looked at in greater depth.

SUBJECT: IB MATHEMATICS SL                AARON SOLENDER

Stellar Numbers

Objective

The objective of this Internal Assessment is to investigate the geometric numbers which lead to special numbers.  The simplest example of these is square numbers, but over the course of this investigation, both triangular numbers and stellar numbers will be looked at in greater depth.  The investigation is conducted for the International Baccalaureate Mathematics Standard Level class and has the purpose of looking at this mathematical concept at a much higher level. The assessment can be broken down into two major sections.

1. Triangular Numbers
1. Triangular Numbers are investigated to begin this assessment in order to give a general background to the concept and idea of special numbers, as to set a background for the in-depth look at stellar numbers.
1. Stellar Numbers
1. Stellar Numbers are the investigation itself, and over the course of the problem, expressions for stellar numbers at different stages will be found, along with general statements for stellar numbers with different amounts of vertices, tests for the validity of these general statements, scopes and limitations of these general statements, and explanations of how arrivals at these general statements were made.

Materials

• The following materials were used over the course of this Internal Assessment
• Texas Instruments TI-83 Graphing Calculator
• Microsoft Word ’98 and Microsoft Equation Editor

Question 1

1. Complete the triangular numbers sequence with three more terms.

The given diagrams for the problem were  through  and are shown below.

The next three terms needed to complete the diagram are  through . These are illustrated in the pictures below.  The white lines on bottom of each diagram are the dots that need to be added to go from one triangular number to the next.

Question 2

Find a general statement that represents the nth triangular number in terms of n

When completing different trials in attempts to find the general term for Triangular Numbers, the following information was determined.

S1=1

S2=3

S3=6

S4=10

S5=15

S6=21

S7=28

S8=36

S9=45

S10=55

The first pattern that I noticed when looking at this information was that the order looks like this when broken down:

1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, … , 1+2+3+4+5+6+7+8+9+10

The first attempt I made was saying the general term was simply (n+1).  After testing the validity of this statement by plugging in different numbers, it was very clear that this was incorrect.

The second attempt made was started off by determining that the first term could be set as equal to .    To continue on this trend, I ...