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

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

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.




Math 24 Fall 2001 2001-09-24