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...

...The aim of this task is to investigate 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 is always added to the previous as shown below, a triangular number will always be the outcome:
1 = 1
2 + 1 = 3
3 + (2 + 1) = 6
4 + (1 + 2 + 3) = 10
5 + (1 + 2 + 3 + 4) = 15
Moreover, triangular numbers can be seen in other mathematical theories, such as Pascal’s triangle, as shown in the diagram below. The triangular numbers are found in the third diagonal, as highlighted in red.
The first diagrams to be considered show a triangular pattern of evenly spaced dots, and the number of dots within each diagram represents a triangular number.
Thereafter, the sequence was to be developed into the next three terms as shown below.
The information from the diagrams above is represented in the table below.
Term Number (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Triangular Number (Tn) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 |
Establishing...

...
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 other graphic packages to find the general pattern to support their result
The general term in this stage is
2. 6- stellarnumbers
Stage
No of dots
1
1
2
13
3
37
4
73
5
121
6
181
Students are expected to make the stellar shape for the next to stages and count the no of dots to get the 6-stellarnumber in 5th and 6th stage
Diagrams can be hand made or using technology
Communication or observation of the number pattern has to be given
From the observation, the expression of the terms of this sequence has to be identified
Expression for the 7th term
General expression 6 – stellar shape
or
p n (n-1) +1
Other stellar shapes – based on the no of vertices the students choose...

...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, 28, and 36.
A general statement for nth triangular numbers in terms of n is:
The 6-stellar star, where there are 6 vertices, has its first four shapes shown below:
The number of dots until stage S6: 1, 13, 37, 73, 121, 181
Number of dots at stage 7: 253
Expression for number of dots at stage 7:
Since the general trend is adding the next multiple of 12 (12, 24, 36, 48 etc…) for each of the stars, so for S2 it would be 1+12=13, and for S3 it would be 13+24=37
General statement for 6-stellar star number at stage Sn in terms of n:
For P=9:
Since S1 must equal 1 then we can prove this formula by showing that:
So the first six terms are: 1, 19, 55, 109, 181, 271
Therefore the equation for the 9-Stellar star at
For P=5:
Since S1 must equal 1 then we can prove this formula by showing that:
So the first six terms are: 1, 11, 31, 61, 101, 151
So the...

...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 triangular grid of evenly spaced dots.
The sequence of triangular numbers is shown in the diagrams above. The first stage has 1 dot; the second stage has 3 dots (1+2); the third stage has 6 dots (1+2+3); the fourth stage has 10 dots (1+2+3+4); the fifth stage has 15 dots(1+2+3+4+5); the sixth stage has 21 dots (1+2+3+4+5+6) ; the seventh stage has 28 dots(1+2+3+4+5+6+7) and the eighth stage has36 dots(1+2+3+4+5+6+7+8). As it could be noticed, there is a sequence where in every stage the number of dots is obtained by adding up all the positive integers that correspond to the previous stages and every time one more number is added.
In terms of n, where n matches up to the stage number, it is accurate to establish an equation so that when trying to find the number of dots in stage 592, it is easy and fast by simply applying the following formulae:
Now it is possible to find the nth number by using the...

...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 |
|3 |9 |
|4 |16 |
In the table on the left, I observe that from the y value 1 to the y value 4 there is an increase of 3. From the y values 4 to 9, there is an increase of 5. From the y values 9 to 16, there is an increase of 7. This shows that it goes: +3, +5, +7, which is then increasing by 2 between each of those numbers.
Below, is the graph of y=x2
[pic]
The equation y=x2 comes from the general equation y= ax2-bx+c.
Y=x2 is the same as y=x2+0x+0. Therefore, a=1, b=0, and c=0.
The next example I am going to show you is similar to the one above. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, 10, 15, 21, 28, 36).
|Tn |y |
|1 |1...

...– 6 – MATME/PF/M11/N11/M12/N12
For 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 (1,3,6,)….
1 3 6 10 15
Complete the triangular numbers sequence with three more terms.
Find a general statement that represents the nth triangular number in terms of n.
Consider stellar (star) shapes with p vertices, leading to p-stellarnumbers. The first four representations for a star with six vertices are shown in the four stages S1–S4 below. The 6-stellarnumber at each stage is the total number of dots in the diagram.
S1 S2 S3 S4
Find the number of dots (i.e. the stellarnumber) in each stage up to S6. Organize the data so that you can recognize and describe any patterns.
Find an expression for the 6-stellarnumber at stage S7.
Find a general statement for the 6-stellarnumber at stage Sn in terms of n.
Now repeat the steps above...

...
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 state their case on the round table of a meaningful contribution to the organisation. This is true in my opnion as in the current day environment more and more is experted and demanded from internal audit fucntion that ever before over and above the traditional mundane internal audit activities.
Definition of internal audit which states that
“Internal auditing is an independent, objective assurance and consulting activity designed to add value and improve an organisation’s operations. It helps an organisation accomplish its objectives by bringing a systematic, disciplined approach to evaluate and improve the effectiveness of risk management, control and governance processes” (Institute of Internal Auditors, 2014). Drawing from this broad definition on can tell that the internal audit function has a serious case of communication it needs to do in order to achieve its objective of adding value and improving the organisations operations. It needs to communicate the results of its...

...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 identity as a largely rural country. Thirty years ago, when China started modernizing its economy, more than 80% of Chinese lived in the countryside. And just six years ago it still was about 60%. Today China is just under 50% urban.
The newly urbanized population will live in eight megacities, those with a population of more than 10 million, as well 15 big cities with populations between 5 million and 10 million. In addition, by 2025 China will probably have at least 221 cities with a population over 1 million, estimates Woetzel. That compares with 35 cities of that scale across all of Europe today. These new urbanites are expected to be a powerful booster of growth: Urban consumption as a share of gross domestic product will most likely rise from 25% today to roughly 33% by 2025. “Urbanization is the engine of the Chinese economy—it is what has driven productivity growth over the last 20 years,” says Woetzel. “And China has the potential to keep doing this for the next 20 years.”...