- Topics At a Glance
- Sequences
- Defining Sequences and Evaluating Terms
- Patterns
- Sequences Can Start at
*n*= 0 **Special Types of Sequences**- Arithmetic Sequences
**Geometric Sequences**- Comparing Arithmetic and Geometric Sequences
- Visualizing Sequences
- 2-D Graphs
- Convergence and Divergence of Sequences
- Other Useful Sequence Words
- Heads and Tails
- Word Problems
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We already know that an arithmetic sequence is one where the difference between successive terms is constant. The distance each term is the same. A geometric sequence is a lot like an arithmetic sequence, but it is completely different at the same time. We can think of it as the doppelgänger of the arithmetic sequence, if we like.

In a **geometric sequence**, the *ratio* between successive terms is constant. Geometric sequences grow or shrink at the same ratio from one term to the next. If we divide any two consecutive terms, we will always get the same ratio.

So if we divide the first terms in a geometric sequence of burritos and get a pinto bean, then dividing burrito-terms two and three of the geometric series will give us a pinto bean of the same size. Let's hope we all like pinto beans.

Is the sequence

2,4,8,16,...

geometric?

Answer.

Look at the ratios of successive terms:

Since these ratios are all the same, the sequence is geometric.

Is the sequence

5,10,15,20,...

geometric?

Answer.

Look at the ratios of successive terms:

We don't have to go any farther. The ratios are not all the same, so this sequence is **not** geometric.

Mom always said we should try to keep a positive attitude, but it's okay for the ratio of a geometric series to be negative. The ratio can be anything nonzero.

When the ratio is negative, successive terms change signs, kind of like the flip-flopping snap of a flag waving in the breeze, always pointing one direction and then the other.

Is the sequence

3,-6,12,-24,...

geometric?

Answer. Let's look at the ratios between successive terms:

Since the ratios are all -2, this IS a geometric sequence.

We mentioned before that an arithmetic sequence is usually defined in terms of subtraction, we can also think of it in terms of addition. To get from one term to the next, you always add the same thing.

Similarly, while a geometric sequence is defined in terms of division, we can also think of it in terms of multiplication. We said that in a geometric sequence, to get from one term to the previous term, you always divide by the same thing. This is the same as saying that to get from one term to the next, you always multiply by the same thing. This "thing" isn't a swamp monster that has come from the deep to scare little children on Halloween. It is called the **common ratio** of the sequence and is usually denoted *r*.

As with the arithmetic sequences, we usually start the geometric sequences off at *a*_{1}. If we know *a*_{1} and *r*, we know everything there is to know about a geometric sequence. We are the masters of the geometric sequence, and maybe the swamp monster too.

A geometric sequence has *a*_{1} = 4 and *r* = 2. Find the first four terms of the sequence.

Answer.

We're given *a*_{1}. To get the next term, we multiply the term we're on by *r* = 2.

*a*_{2} = *a*_{1}(2) = 8

*a*_{3} = *a*_{2}(2) = 16

*a*_{4} = *a*_{3}(2) = 32

Like eating a piece of cake, we figured out the general formula on our own. To get from *a*_{1} to *a _{n}* we have to multiply by

(If it's not quite clear why it's (*n* – 1) times, go read about steps in arithmetic sequences).

Multiplying by *r* a total of (*n* – 1) times is the same thing as multiplying by *r*^{n – 1}, by the definition of what exponents are.

If we start at *a*_{1} and multiply by *r*^{n – 1}, we end up at

*a _{n}* =

Because mathematicians need to save their writing hand for scribing infinite sequence strings, it's common to abbreviate* a*_{1} by *a*. so you'll probably see this formula written

*a _{n}* =

In everything we do with geometric sequences from here on, the letter *a* refers to the *first term of the sequence*.

Now that we have this nice formula, if we know *a* and *r* for a geometric sequence, we can easily find any term we like.

Just like an arithmetic sequence, if we're given two consecutive terms of a geometric sequence we can work backward to find *r* and then *a*. We know that *r* is the ratio of successive terms:

Once we know *r* we can use the formula

*a _{n}* =

to find *a*.

Just when we thought we had it all figured out, we have a new problem. How do we find *r* and *a* if we have two non-consecutive terms *a _{m}* and

This is no impossible labyrinth to find our way out of. We only need to perform one extra operation to solve this problem, but it requires knowing how many times we multiply by *r* to get from *a _{m}* to

Let's look at our equation for a geometric sequence. The *n*^{th} term is

*a _{n}* =

while the* m*^{th} term is

*a _{m}* =

Let's divide these equations:

We can cancel the <em>a</em> out, and we will multiply by , which is just 1 in disguise. Sneaky devil. This gives us

So we get

So, once we know how many steps we need to take, we have (*n* – *m*). We divide *a _{n}* by

1) Find .

2) Take the (*n* – *m*)th root to get *r*.

3) Use the geometric sequence formula to find *a*.

Remember that we should always check our answers by using the values of *a* and *r* you found in the formula. If we found the right values, we should be able to use the formula to get the terms *a _{m}* and

As with an arithmetic sequence, a geometric sequence is completely determined by two pieces of information. If you know *a* and *r*, you know everything there is to know about the sequence. If you know any two terms *a _{m}* and

Example 1

A geometric sequence has |

Example 2

A geometric sequence has (a) (b) (c) (d) |

Example 3

A geometric sequence has |

Example 4

A geometric sequence has |

Example 5

Find |

Example 6

Find |

Example 7

Find |

Exercise 1

Determine whether the sequence is geometric.

1,

Exercise 2

Determine whether the sequence is geometric.

3,6,9,12,...

Exercise 3

Determine whether the sequence is geometric.

10, 15, 22.5, 33.75, ...

Exercise 4

Determine whether the sequence is geometric.

3, 12, 48, 192, ...

Exercise 5

Determine whether the sequence is geometric.

.9, .8, .7, .6, ...

Exercise 6

Determine if the sequence is geometric or not.

1,-1,1,-1,...

Exercise 7

Determine if the sequence is geometric or not.

-2,14,-98, 686, ...

Exercise 8

Determine if the sequence is geometric or not.

3, -15, -75, 375, ...

Exercise 9

Find the first four terms of the geometric sequence with

*a*_{1} = 7 and *r* = 3

Exercise 10

Find the first four terms of the geometric sequence with

*a*_{1} = 100 and

Exercise 11

Find the first four terms of the geometric sequence with

*a*_{1} = 2 and *r* = 0.5

Exercise 12

Find the first four terms of the geometric sequence with

*a*_{1} = 8 and *r* = -4

Exercise 13

Find the first four terms of the geometric sequence with

*a*_{1} = 3 and *r* = 1

Exercise 14

A geometric sequence has *a*_{1} = 4 and *r* = 5. Find

a) *a*_{2}

b) *a*_{3}

c) *a*_{4}

d) *a _{n}*

Exercise 15

A geometric sequence has *a*_{1} = 3 and . Find

a) *a*_{2}

b) *a*_{3}

c) *a*_{4}

d) *a _{n}*

Exercise 16

A general geometric sequence has first term *a*_{1} and common ratio *r*. Find

a) *a*_{2}

b) *a*_{3}

c) *a*_{4}

d) *a _{n}*, the general formula for the

Exercise 17

A geometric sequence has *a* = 8 and *r* = 0.2. Find *a*_{5}.

Exercise 18

A geometric sequence has *a* = 2 and *r* = -5. Find *a*_{7}.

Exercise 19

A geometric sequence has *a* = 0.125 and *r* = 2. Find *a*_{20}.

Exercise 20

Find *a* and *r* for the geometric sequence with the given terms.

*a*_{7} = 640 and *a*_{8} = -1280

Exercise 21

Find *a* and *r* for the geometric sequence with the given terms.

*a*_{4} = 1.2 and *a*_{5} = 0.24

Exercise 22

Find *a* and *r* for the geometric sequence with the given terms.

*a*_{4} = -3.125 and *a*_{5} = 0.78125

Exercise 23

Find *a* and *r* for the geometric sequence with the given terms.

*a*_{11} = 1 and *a*_{13} = 0.0625

Exercise 24

Find *a* and *r* for the geometric sequence with the given terms.

*a*_{4} = 51.2 and *a*_{7} = 26.2144