# Chapter 1Edit

## Section 2Edit

### 1Edit

by associativity | ||

by commutativity | ||

by commutativity | ||

by associativity |

### 14Edit

### 15Edit

multiply both sides by -1. |

## Section 3Edit

### 30, 31, 32, 33Edit

Although these can be done by hand the exercises suggest that you derive a general formula for finding the final population figures. The following one will suffice for the stated problems, where is the initial population, is the scaling factor (doubles, triples, etc.), and is the inital year and end year respectively, and is the number of years taken for the population to go up by :

For example, 32a:

## Section 4Edit

### 8-15Edit

Write out the powers of 2 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.

### 16-23Edit

Write out the powers of 3 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.

### 24Edit

#### a)Edit

Examining the definitions first, if and then and .

means that .

Let

#### b)Edit

Examine the definitions in part a) again. We have and .

Solving for and :

and

Let .

.

## Section 5Edit

### 7Edit

#### a)Edit

120, 720, 5040 and 40320.

#### c)Edit

,

,

#### d)Edit

Mutliply both sides of the equation by and , cancelling unecessary factors to 1. We do this in order to achieve the denominator :

Use the fact that and to achieve:

Hence,

## Section 6Edit

### 1Edit

#### a)Edit

### 2Edit

#### a)Edit

#### b)Edit

Recall from the chapter that if , then .