# Sequences, Probability and Counting Theory

## Geometric Sequences


### Learning Objectives

1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).

### Objective 1: Determine if a sequence is geometric (IA 12.3.1)

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same.

The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.


### Practice Makes Perfect


Determine if each sequence is geometric. If so, indicate the common ratio.



### Practice Makes Perfect


### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)

Let’s find the formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is  and the common ratio is  . We will then look for a pattern.


### Practice Makes Perfect


Find the general term (nth term) of a geometric sequence.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

### Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.



### Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is 
  and the common ratio is 
  we can find subsequent terms by multiplying 
  to get 
  then multiplying the result 
  to get 
  and so on.
The first four terms are 
 


### Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

### Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence 
  This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .

### Solving Application Problems with Geometric Sequences

In real-world scenarios involving geometric sequences, we may need to use an initial term of 
  instead of 
  In these problems, we can alter the explicit formula slightly by using the following formula:

### Key Equations


### Key Concepts

1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See  and .
5. A recursive formula for a geometric sequence with common ratio 
  is given by 
  for 
  .
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio 
  is given by 
  See .
8. In application problems, we sometimes alter the explicit formula slightly to 
  See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.

### Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

### Extensions 
