Partial Fraction Decomposition

For partial fraction decomposition, your either going to be solving it as a linear or quadratic problem. The steps include first multiply by the LCD. Second group terms by powers of x. Lastly, equate coefficients and solve the system of equations. If the degree of the numerator is bigger than the degree of the denominator then we will be using long division. And when it's a quadratic problem, we write Bx+C as the numerator. 

Probability of an Event

In probability theory, an event is a set of outcomes of an experiment, to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. In words, this means that the probability of an event must be a number between 0 and 1(inclusive). Also, probability of an impossible event is always zero. On the other hand, the probability of an impossible event is always zero. 

Recursion and mathematical Induction

In the Tower of Hanoi Puzzle, I learned how many different possible methods of moving the rings around. I decided to move the first ring to the middle, so that I could get the other rings to the far side. However, it became difficult because the rings had to be stacked from greatest on the bottom to the smallest on the top. Because of this, I couldn't stack the third ring first and then the fourth on top. I had to make smart choices of where I would place each ring because there were many obstacles and rules in the puzzle itself. I also learned many things from recursion and mathematical induction such as that n always equals one and then we must assume that the statement is held true for n equals k, and also prove it true for k+1. When solving the proof for the Tower of Hanoi Puzzle, the general equation was Tn=2 to the n exponent minus one. The first step was that I had to show this true for n=1 and I had to explain why this was true. The second step was that I assumed true for n=k and I substituted k in for n. Lastly I had to show true for Tk+1 and plug in k+1 for n in the equation. I was able to solve and prove the proof was correct. 

Sequences and Series

A sequence can be described as a list of objects, events, or numbers that come one after the other, that is, a list of things given in some definite order. Each object in the list is called a term of the sequence. Where no last term is indicated, is understood to be an infinite sequence. The three dots is called an ellipsis amd indicates that succeeding terms follow the same pattern as that set by the terms given. The elements in the range of a sequence are simply the terms of the sequence. The nth term f(n)=an is also called the general term of the sequence. 

Graphing Systems of Inequalities

To graph systems of inequalities there are certain rules that must be followed. First, solid lines are used when the inequality could be equal to, dashed lines are for strict inequality. Second, when a line is in slope-intercept form, shading is above the line when it is greater than and the shading is below the line when it is lesser than. When a line is in standard form, the shading is the same as it is in slope-intercept form if the coefficient of the y-variable is positive, otherwise it is reversed. When working with standard form, it is sometimes necessary to choose better numbers to ensure the graph is precise.