Homework - September 26, 2001

Topics Covered: Principle of Mathematical Induction, proofs by contradiction, vectors spaces (Sec. 2.1) Reading Assignment:

1. Appendix C - Fields pp. 510-513,
2. Handout $\char93 1$
3. Sect. 1.2 and 1.3

Homework Problems:

1.

In no more than one page write a summary of what is induction and why it works as a method of proof. Assume your audience is someone with very little mathematical sophistication.

2.

Use induction to prove that

$$\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}$$

3.

Use induction to prove that $(1+x)^n\geq 1+nx$ if $x\geq 0$ and $n$ is a positive integer.

4.

Find an error in the following inductive "proof" that all positive integers $n$ are equal. Let $S$ be the set of all $n$ such that $n$ equals all integers between 1 and $n$. Then $1\in S$. Now suppose all integers up to and including $k$ are in $S$. Then $k=k-1$, so adding 1 to both sides gives that $k=k+1$. Therefore, by the principle of mathematical induction, $S$ contains all positive integers, and so all positive integers are equal.

5.

Use the strong form of induction to prove that any integer $n\geq 24$ can be expressed as $n=5x+7y$, where $x$ and $y$ are nonnegative integers.

6.

Sect. 1.2 $\char93$ 1,10,13,18, 22.