Patterns and sequences are the basis of mathematical understanding. Based on geometry alone, many special patterns evolve, such as the square numbers, triangular numbers, and much more. The Stellar Numbers are mostly used in astronomy and astrology. Stellar Numbers are figurate numbers based on the number of dots that can fit into a midpoint to form a star shape. The points of the star determine the number of points plotted around the midpoint. Triangular numbers is a figurate number system that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. The triangular numbers from that pattern are 1 followed by 1+2 followed by 1+2+3 and so on. From the pattern of the triangular numbers, this infinite serious starts with 1, 3, 6, 10, 15… With this pattern, calculated by counting, the next three terms would be 21, 28, and 36. To derive a formula from this pattern, we can see that x is repeated and the number goes up each time by one. After using the rules of the sequences and a few checks, the final formula results inx(x+1)2 where x is any natural number. I found this formula with the calculator with steps show below. To prove this formula, there is the typical guess and check formula where the number of dots in the next triangle is counted for. Strangely when noticed clearly, the triangular numbers can be found in the third diagonal of Pascal’s triangle, starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the forth is 10, and so on. This following drawing below shows the triangular pattern of evenly space dots. In this first picture, the triangular numbers with three more terms is completed.

Solution One: Technology
The next step is to find a general statement that represents the nth number of dots in the triangular number series in terms of x. Let X represent the stage and let Y represent the corresponding triangular number: x| 1| 2| 3| 4| 5| 6| 7| 8|

y| 1| 3| 6| 10| 15| 21| 28| 36|

The calculator used in the production of this formula was Texas Instrument TI-84 Plus Silver Edition. Select STAT. Punch in the value of X in list 1 and the values of Y in list 2. Select CALC 5 to Quadratic Regression to get y=ax2+bx+c where a, b=0.5 and c=0. y=0.5x2+0.5x

=0.5(x2+x)
=x(x+1)2 General Statement

Solution Two: Caveman
To derive a formula in which X and Y are related to find the next terms, it will either be a geometric sequence, arithmetic sequence, or neither. In this case, it is neither arithmetic sequence nor a geometric sequence because the ratios of each term in Y do not equal and the values of differences are not the same. So there has to be another way to derive the formula. From calculations, differences between two adjacent terms have a clear pattern/arithmetic sequence.

If the left side of all the calculations are added together, all the terms except Yx canceled because of the positive and negatives. The right side of the calculations adds up is the sum of the arithmetic sequence from 1 to x. In this manner, we can easily calculate the relation between Y and X. (Details following)

Task: to find Y in terms of X.
Y1 =1
Y2-Y1=2
Y3-Y2=3
Y4-Y3=4

Yx-1-Yx-2=x-1
Yx-Yx-1=x
Y1 + Y2-Y1 + Y3-Y2 + Y4-Y3 …. + Yx-1-Yx-2 + Yx-Yx-1 = 1 + 2 +3 +4 + … +x If the left side of all the calculations are added together, all the terms except Yx canceled because of the positive and negatives. The right side of the calculations adds up is the sum of the arithmetic sequence from 1 to x. In this manner, we can easily calculate the relation between Y and X.

Yx = 1+2+3+ 4 + … + x
= x(Y1+Yx)2
= x(x+1)2 General Statement...

...geometric shapes, which lead to special numbers. The simplest example of these are square numbers, such as 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3, and 4.
Triangular numbers are defined as “the number of dots in an equilateral triangle uniformly filled with dots”. The sequence of triangular numbers are derived from all natural numbers and zero, if the following number...

...
StellarNumbers
Results
1. Triangular Numbers
Observation of the number pattern of polynomial type or different pattern needed.
Identifying the order of the general term by using the difference between the succeeding numbers.
Students are expected to use mathematical way of deriving the general term for the sequence.
Students are expected use technology GDC to generate the 7th and 8th terms also can use...

...Math SL Investigation Type 2
StellarNumbers
This is an investigation about stellarnumbers, it involves geometric shapes which form special number patterns. The simplest of these is that of the square numbers (1, 4, 9, 16, 25 etc…)
The diagram below shows the stellar triangular numbers until the 6th triangle.
The next three numbers after T5 would be: 21,...

...STELLARNUMBERS
In order to develop this mathematics SL portfolio, I will require the use of windows paint 2010 and the graphic calculator fx-9860G SD emulator, meaning that I will use screenshots from this software with the intention of demonstrating my work and process of stellarnumbers sequences.
Triangular numbers are those which follow a triangular pattern, these numbers can be represented in a...

...Introduction: In this following assignment, I will be considering geometric shapes that lead to special numbers. The simplest examples of these are square numbers (1, 4, 9, 16, etc), which are derived from squaring 1, 2, 3, and 4. From this I got the equation y= x2. This equation is illustrated in the table below.
y=x2
|x |y |
|1 |1 |
|2 |4...

...final assessment in 2011 and 2012
STELLARNUMBERS SL TYPE I
Aim: In this task you will consider geometric shapes which lead to special numbers. The simplest example of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4.
The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers...

...
MPhil Internal Auditing
IOK 821 – Communication Management
Student No:xxxxxx
Individual Assignment: Increasing the relevance/value of the internal audit function
.
Table of Contents
1. Introduction
In this assignment will focus on how to increase the relevance/ value of the internal audit function in an organisation through the various aspects of communication.
Drent ( 2002), states that on daily basis internal auditors are faced with challenage to...

...The numbers are overwhelming: Over the next 17 years, 350 million rural residents (more than the entire U.S. population today) will leave the farm and move to China’s cities. That will bring the Chinese urban population from just under 600 million today to close to 1 billion, changing China into a country where more than two-thirds of its people are city dwellers, says Jonathan Woetzel, a director in McKinsey’s Shanghai office. The change will reverse China’s centuries-old...