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title: MTH 350 Homework 1
tags: [' homework', mth350]

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tags: mth350, homework
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# MTH 350 Homework 1

Instructions for this Homework set are on Blackboard where this is posted. 

## Problem 1

:::info
This problem is an **independent** problem. As described in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (top of page 14) this means that **on this problem, the only help you can get is from the professor.** 
:::

**Prove Theorem 1.1.7**: Given a real number $b > -1$, $(1+b)^n \geq 1 + bn$ for all $n \in \mathbb{N}_0$. 

(This is the problem whose proof framework we did in class on January 13.)

## Problem 2

:::info
This problem is a **collaborative** problem. This means that **you can work on it in a small group, as long as you stay within the bounds of academic honesty** laid out in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (starting on page 13). 
:::

**Choose exactly one of the following statements** and write a complete, correct, and clear proof using mathematical induction. (Even if you can think of a way to prove it without induction, use induction here since that's the focus of this homework.) 

You can (and should) test-drive each of these problems in your notes to see which one feels best for you, but please only turn in work on one of these (in addition to Problem 1). 


1. Suppose $A_1, A_2, A_3, 
\dots, A_n$ and $B$ are sets. Prove that for all $n \in \mathbb{N}$,   
$$(A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n) \cup B = (A_1 \cup B) \cap (A_2 \cup B) \cap (A_3 \cup B) \cap \dots \cap (A_n \cup B)$$
2. Prove that for every $n \in \mathbb{N}_0$, a set with $n$ elements has $2^n$ subsets. (Reminder: If $A$ is a set, then both the empty set $\emptyset$ and $A$ itself are subsets of $A$.)
3. Prove that for each $n \in \mathbb{N}$, 
$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots + \frac{1}{(2n-1)(2n+1)} = \frac{n}{2n+1}$$
3. Prove that for every natural number $n$, $\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \dots + \binom{n}{n} = 2^n$. The expression $\binom{n}{k}$ is the *binomial coefficient* "$n$ choose $k$". (If you don't know what this is... you can either learn it on the fly, or just don't do this one!)
3. Prove that for all $n \in \mathbb{N}$, 
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^n} = 1 - \frac{1}{2^n}$$

