...Originally Pascal’s Triangle was developed by the Chinese of long ago. But then the French mathematician Blaise Pascal was officially the first person to discover the importance of the patterns it had within itself. But how exactly does it work??? In this research paper, I will explain how to make the Pascal’s Triangle and why it is so special.
Construction:
Pascal’s Triangle is basically a triangle of numbers. “At the tip of the triangle is the number 1, which makes up row zero. Then the second row has two 1’s by adding the 2 numbers above them to the left and right, 1 and 0 (all numbers outside the triangle are zeros). Now do the same for the second row.” 0+1= 1, 1+2=3, 2+1=3, 1+0=1. Then the results become the third row. 0+1=1, 1+3=4, 3+3=6, 3+1=4, and 1+0=1. Then the pattern continues on infinitely.
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There seems to be many patterns in this triangle. For example: The Sums of the Rows.
The Sums of the Rows:
“The sum of the numbers in any row is equal to 2 to the nth power or 2^n when n represents the number of the row.” For example:
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
Prime Numbers:
“If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21,...

...Pascal’s Triangle
The Pascal’s Triangle is a triangular array of the binomial coefficients. The system after French mathematician Blaise Pascal. The set of numbers that form Pascal's triangle were known before Pascal. However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers.
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala in or before the 2nd century BC.While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who realized the combinatorial significance.
At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029).It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám...

...Name: Linh Nguyen
IB MathIA
02/06/12
Part A
Consider this 2× 2 system of linear equations: x + 2y = 3
2x - y = -4
We can see patterns in the constants of both equations. In the first equation, the constants are 1, 2, and 3. The common difference between the constants is 1:
3 – 2 = 1
2 – 1 = 1
Based on this, we can set up a general formula for the constant of this equation:
Un = U1 + (n - 1)d Where:
n: The number of the series
d: the common difference in the series.
For the second equation, we also can see that the constants belong to a arithmetic series, which has the common difference of -3:
-1 – 2 = -3
-4 – (-1) = -3
Un = U1 + (n+3)d
Solving the equation:
x + 2y = 3
2x – y = -4
* 2x + 4y = 6
2x – y = -4
* 5y = 10
* y = 2
* x + 4 = 3
* x = -1
Looking at the graph, we can see the intersection point A has the coordinate (-1, 2). At this point, the two lines are equal.
Solving equation with similar formats:
x + 6y = 11 Common difference is 5
6x – 2y = -10 Common difference is -8
* 6x + 36y = 66
* 38y = 76
* y = 2
* x = -1
6x + 7y = 8 Common difference is 1
2x + 4y = 6 Common difference is 2
* 6x + 12y = 18
* 5y = 10
* y = 2
* x = -1
Looking at these systems of linear equations, we can conclude that any equations which have their constants follow a arithmetic series will have the answer for the variable y as 2 and variable x as -1....

...IB Math Studies Internal Assessment:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Exam Session: May 2014
School Name:
Teacher:
Course: IB Math Studies
Word Count: 654
Name:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Introduction
In tennis, players hit the tennis ball in certain ways so the ball goes the way they want it to go. Hitting it at certain angles enables the tennis ball to travel various distances, it depending on the angle. Some angles make the ball go short or far distances.
Statement of Task
The main purpose of this investigation is to determine whether there is a relationship between the distance a tennis ball travels horizontally and the angles of which it is dropped at. The type of data that will be collected is that I will drop a tennis ball, at a constant height of 4 feet, at from different angles (30°, 40°, 50°, 60°, 70°), twenty times for each angle with a total of one-hundred drops and measuring the distance of each drop in feet. The data will be used to determine whether the angle of which a ball is dropped affects how far a tennis ball travels horizontally.
Plan of Investigation
I am investigating if the distance a tennis ball travels horizontally dependent on the angle it is drops at. I have collected data on the distances a tennis ball travels...

...MathIA
IB MATH SL
MathIA
Introduction: In this task I will consider a set of numbers that are presented in a symmetrical pattern and try to find a general equation to find the elements in the [pic]row.
Consider the five rows of number shown below.
Figure 1 Lacsap’s Fractions
The aim of this task is to find the numerator of the sixth row and to find the general statement for [pic]. Let [pic] be the [pic]element in the [pic]row, starting with r=0.
First, I will make a table of the numerator and the row number to show the relationship between the numerator and the row number.
Table 1 A table showing the relationship of rows and numerator
The difference between the numerator of row 1 and row 2 is 2, then the difference between the numerator of row 2 and row 3 is 3. The second difference for each row is 1 and it shows that this is a geometric sequence. So, I will start by finding the equation using the quadratic formula, [pic], in which x is the row number and y is the numerator.
First, I will plug in the numbers in the second row, which is 2 for x and 3 for y to try finding the quadratic equation of the Lacsap’s sequence. Then, I will plug in the numbers in the third row and it forms a simultaneous equation. I will use substitution method to solve the equation as there are two unknowns in the...

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ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which resists relative lateral motion of two solid surfaces in contact and is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces; lubricated friction which is a case of fluid friction where a fluid separates two solid surfaces; skin friction which is a component of drag, the force resisting the motion of a fluid across the surface of a body; internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation and sliding friction.
When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into heat. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to heat whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear,...

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The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.
A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations. An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted. It has to be clear and certain in terms. It must also be communicated to the offeree before it is being accepted. In addition, the acceptance has to be unqualified, unconditional and made by a positive act. In the case of Beauty and Stylish, a positive act refers to the signing of the contract. All terms of the offer must be accepted without any changes and cannot be subjected to any condition, taking effect only upon fulfillment of that condition. When Beauty and Stylish enter into the agreement, they must intend to bind and bound legally to each other by their agreement. This is the intention to create legal relations between two parties. In the meanwhile, this contract must possess consideration. A contract must therefore be a two-sided affair, with each side providing or promising to provide something of value in exchange for what the other is to provide.
Every contract, whether oral or written, contain terms. The terms of a contract set out the rights and duties of the parties. Terms are the promises and undertakings given by each...

...• What were the most revolutionary social and economic developments of the last quarter of the nineteenth century?
• How did different groups of Americans respond to those changes and how effective were their responses?
• What role did government play in these developments?
In the late 1900s some of the most social and economic developments were railroads, steel oil, the type writer cash register, light bulb and agriculture. The development of the railroad made it easier for immigrant to come to this country for work. This meant that there were more group of different races and cultures in America. And in some states there became an over population and city workers like police and garbage men could not keep up with the demand of so many people. Some groups mover to open land for the Homestead Act. They had hope of farming and staying on the land for at least five years as agreed but the supply and demand of agriculture did sustain so many farmer moved off the land well before their five years. The government played many different role I deescalating some issues in American history. Women and children were being worked for long hours and getting paid a little bit of nothing in return for their hard work. So the government put labor laws into place that were to protect women and children. As oil, steel and railroad industries grew so did the levels of pollution. The government again put laws that were to protect animals and the earth so that there would not...