Tower of Hanoi-Mathematical Induction

The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower,[1] and sometimes pluralized) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules: Only one disk can be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack. No disk may be placed on top of a smaller disk. With three disks, the puzzle can be solved in seven moves. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks.

But you cannot place a larger disk onto a smaller disk. The equation for the tower of Hanoi is 2^n -1. The equation 2^n-1 can be used to solve with any number of discs. Given 12 discs, this would require 4,095 moves.

As disc continue to stack, more moves are required to move them from pole A to C. I also learned that after finding the recursion formula, I was able to plug it into the mathematical induction to make both sides equal.

2nd Semester Summary

It's been really confusing this semester, but it was so much easier than 1st semester. I feel really sad that now I will no longer be with Ms. V. And now I might have to move up to Calc AB and be with Pelletier. I am so scared because my grades were not that grades this year and so that might affect next year, especially knowing how tough Ms. Pelletier might be. D:

Trig Review

In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine. sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) 1 + cot2(t) = csc2(t) The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1. sin(–t) = –sin(t) cos(–t) = cos(t) tan(–t) = –tan(t) Notice in particular that sine and tangent are odd functions, while cosine is an even function.

Repeating Decimals

A repeating or recurring decimal is a way of representing rational numbers in base 10 arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero. For example, the decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333…. A more complicated example is 3227 / 555 , whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144…. At present, there is no single universally accepted notation or phrasing for repeating decimals.

Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats but extends forever without regular repetition. Examples of such irrational numbers are the square root of 2 and pi.

Parametric Equations

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. are parametric equations for the unit circle, where t is the parameter. 

Together, these equations are called a parametric representation of the curve. A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter. The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.)

Partial Fraction Decomposition 

Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x(x + 1). Then you write the fractions with one of the factors for each of the denominators. Of course, you don't know what the numerators are yet, so you assign variables (usually capital letters) for these unknown values: A/x + B/(x + 1) Then you set this sum equal to the simplified result: (3x + 2)/[x(x + 1)] = A/x + B/(x + 1) Multiply through by the common denominator of x(x + 1) gets rid of all of the denominators: multiply each term by [x(x + 1)] / 1 3x + 2 = A(x + 1) + B(x)

Sequence and Series

A "sequence" (or "progression", in British English) is an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". A "series" is the value you get when you add up all the terms of a sequence; this value is called the "sum". For instance, "1, 2, 3, 4" is a sequence, with terms "1", "2", "3", and "4"; the corresponding series is the sum "1 + 2 + 3 + 4", and the value of the series is 10. A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. So the second term of a sequnce might be named "a2" (pronounced "ay-sub-two"), and "a12" would designate the twelfth term. Note: Sometimes sequences start with an index of n = 0, so the first term is actually a0. Then the second term would be a1. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Don't assume that every sequence and series will start with an index of n = 1.