Math Portfolio Stellar Numbers. This task is an investigation of geometric numbers, the series they create and the general terms of the stellar numbers.

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Math Type I Portfolio: Stellar Numbers

MayExaminations

This task is an investigation of geometric numbers, the series they create and the general terms of the stellar numbers.

Introduction:

Stellar numbers are numbers which pertain to the shape of stars. The numbers are representative of the number of vertices, which are also equal to the number of sides, of geometric shapes. Some examples of geometric shapes are triangular numbers (ie. shapes with three sides and three definite vertices) and square numbers (ie. shapes with four sides and four definite vertices). Stellar numbers are also geometric shapes and may have innumerable vertices and sides. Stellar number, triangular numbers and square numbers all display certain trends and patterns which may be seen in sequence of specific geometric numbers. The following task aims to identify and investigate the general statements of the various geometric shapes.

Triangular numbers

Triangular numbers are numbers of dots which can be arranged to form an ordered equilateral triangle. The Diagram 1 displays the first eight triangular numbers appear.

Figure 1:

Table 1:

The nth term i.e. the nth triangular number could be represented by the arithmetic series:

1+2+3+...+(n-2)+(n-1)+n

This conclusion is justified because the first term u1=1 and all consecutive terms of the series have a common difference of 1.  The above series is a sum of n terms in arithmetic progression with u1=1 and difference of 1. It can also be called the sum of first ‘n’ natural numbers. Because the progression is arithmetic, the following formula can be used to find the sum of ‘n’ triangular numbers:

 d

Where  = 1 and d= 1

=

=

General term of an nth triangle number =        Where  n

Validation

The general term can be validated by substituting the values of ‘n’ by various natural numbers as follows:

General term =

Substitute n with 2

=

=

=3

The number of dots for ‘n=2’ is 3 which is identical to the value projected in the table of values.

General term =

Substitute n with 8

=

=

=36

The number of dots for ‘n=8’ is 36 which is identical to the value projected in the table of values.

Square Numbers

Figure 2:

Table 2:

When observing the series of dots, a pattern emerges where the difference of the difference between two terms is increasing by 2 each time.  The series is arithmetic because the difference remains constant. The number of dots in the nth square number can be expressed as:

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1+3+5+7+....+(

The following formula can be used to find the sum of the series:

Sn=

Sn =

Sn =

Sn =

Sn= n2       Where n

Therefore the relationship between ‘n’ and the number of dots is represented by the general statement: n2

Validation

Term number 4 → 42 = 16 =number of dots

Term number 2 →22= 4 = number of dots

Stellar numbers:

Working for the 6 stellar number:

Where p=6

Figure 3:

Table 3:

Expression for 3-stellar number at stage S4 = ...

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