Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Beginning Activity Beginning Activity 1: Exploring Statements of the Form $(\mathrm{\forall }n\in \mathbb{N})(P(n))$

A4US

One of the most fundamental sets in mathematics is the set of natural numbers $\mathbb{N}\text{.}$ In this section, we will learn a new proof technique, called mathematical induction, that is often used to prove statements of the form $(\mathrm{\forall }n\in \mathbb{N})(P(n))\text{.}$ In Section 4.2, we will learn how to extend this method to statements of the form $(\mathrm{\forall }n\in T)(P(n))\text{,}$ where $T$ is a certain type of subset of the integers $\mathbb{Z}\text{.}$

For each natural number $n\text{,}$ let $P(n)$ be the following open sentence:

All of the examples that were used should provide evidence that the following proposition is true:

We should keep in mind that no matter how many examples we try, we cannot prove this proposition with a list of examples because we can never check if 4 divides $\left(5^n-1\right)$ for every natural number $n\text{.}$ Mathematical induction will provide a method for proving this proposition.

For another example, for each natural number $n\text{,}$ we now let $Q(n)$ be the following open sentence:

The expression on the left side of the previous equation is the sum of the squares of the first $n$ natural numbers. So when $n=1\text{,}$ the left side of equation (15) is $1^2\text{.}$ When $n=2\text{,}$ the left side of equation (15) is $1^2+2^2\text{.}$

All of the examples we have explored, should indicate the following proposition is true:

For each natural number $n\text{,}$ $1^2+2^2+\cdots +n^2={\displaystyle \frac{n(n+1)(2n+1)}{6}}\text{.}$