Tower of Hanoi

The game The Tower of Hanoi is basically a mathematical game or puzzle that 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. Our team tried the game for several times but always takes more steps to complete it. At the end of this game we find an equation show the pattern of this game and can we complete the game perfectly. The equation we prove out is Tn=2^n -1.  The two steps are to show true for n=1 and assume the statement is true for n=k, and prove true for k+1. After we followed these steps to show that k+1 is true and it follows from mathematical induction that the statement is true for every positive integer. For the recursion and mathematical induction, I learned that it is a form of direct proof, and it is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number.

Sequences and series

 A sequence 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.

Graphing system of inequality

Before learning how to graph system of inequality. We first need to learn the symbols for inequalities: The symbol < means less than. The symbol > means greater than. A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system.