Repeating Decimals #13

Repeating decimals are fun! Every repeating decimal is the sum of an infinite geometric series. The key to finding the answer is to write the decimal all as a quotient of integers. In other words, as a fraction. An example would be...

Given: 0.232323...

First step: write as a geometric series

0.23 + 0.0023 + 0.000023 + 0.00000023 + ...

Second step: find the sum

23/100 + 23/10,000 + 23/1,000,000 + ...

r(the pattern) = 1/100

Use the formula S = a/1-r

S = (23/100)/(1-(1/100)) = (23/100)/(99/100) = 23/99

Parametric Equations #12

The first step to solving a parametric equation is to sketch the graph. Making a table helps. Then eliminate the parameter by using substitution, elimination, and trig identities. When it comes to solving, always eliminate "t." The parameter = t. In every set of equations, an interval is given. A key is to look for a trig identity that only has the sin, cos, or tan in the equation. 

Partial Fractions #11

There are three set up steps for partial fraction decomposition. 1. Divide if it is improper. 2. Factor he denominator. 3. If it is linear, add A,B,C...etc. If it is quadratic, add Ax+B, Cx+D...etc. Then, there are four steps for solving. 1. Multiply by the least common denominator. 2. Group the terms by powers of x. 3. Equate the coefficients. 4. Solve the system of equations. 

Graphing Systems of Inequalities #8

For graphing systems of Inequalities, there are three steps. 1. Graph each equation based upon if it is a line, parabola, or circle. The equation for a line is y=mx+b. The equation for a parabola is y=(x-h)^2+k. The equation for a circle is x^2+y^2=r^2. 2. Pick a test point not on the line. 3. Shade the plane containing the test point, if the test point satisfies the equation, shade the other plane if it does not. 

Sequences and Series #9

A sequence is a particular order in which related events, movements, or things follow each other. A series is a number of things, events, or people of a similar kind or related nature coming one after another. There are two types of series: arithmetic and geometric. Arithmetic is when numbers are added or subtracted. To find the sum of an arithmetic series, use the formula Sn=n(a1+a2/2). To fund the sum of a geometric series, use the formula Sn=a1(1-r^2)/1-r. Geometric is when numbers are multiplied or divided. 

Tower of Hanoi-Mathematical Induction #10

In this project, I learned how to do mathematical induction. It was difficult to get all the numbers from the rings, because my screen kept freezing, but when I did get them, I did not realize how many moves it took to complete each step. I decided to move the first ring to the ring beside it because that was the only available place. For me, this game just seemed to be to put the ring wherever it fit, there was not rhyme or reason. Mathematical induction helped us to prove that the equation we came up with for the project was true, because when we plugged in certain variables, they worked. Step one proved that n=1 which was helpful, because we then knew we were on the right track. In step two, we proved that (k+1) was true, therefore proving that our entire equation was true and proven correct. I learned that the recursion formula can be very helpful in terms of plugging in that set for a longer part of the equation. We know that using mathematical induction can help prove something, because if we plug a 1 into each side and they come out equal, then our equation is correct. 

Cramer's Rule #7

Cramer's Rule is a lot easier to use when finding the determinant of each variable. In order to find each determinant, first find the common determinant. Then, substitute the solutions given for the first column and solve for the determinant. Next, substitute the solutions given for the second column and change the first back to its original numbers and solve for the determinant. To find the determinant of X, put the first determinant you found over the common determinant. To find the determinant of Y, out the second determinant you found over the common determinant.