In this section you. will study sequences in which each term is a multiple of the term preceding it. You will also learn how to find the sum of the corresponding series.

**Objectives**

(1). Find the common ratio of a geometric sequence.

(2). Write terms of a geometric sequence.

(3). Use the formula for the general term of a geometric sequence.

(4). Use the formula for the sum of the first *n* terms of a geometric sequence.

(5). Find the value of an annuity.

(6). Use the formula for the sum of an infinite geometric series.

**Geometric Sequences**

**Figure A** shows a sequence in which the number of squares is increasing. From left to right, the number of squares is 1, 5, 25, 125, and 625. In this sequence, each term after the first, 1, is obtained by multiplying the preceding term by a constant amount, namely 5. This sequence of increasing numbers of squares is an example of a geometric sequence.

**Figure A**. A geometric sequence of squares

Definition of a Geometric Sequence

A geometric sequenceis a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called thecommon ratioof the sequence.

**Geometric Sequences**

Consider the following sequence:

Unlike an arithmetic sequence, these terms do not have a common difference, but there is a simple pattern to the terms. Each term after the first is twice the term preceding it. Such a sequence is called a geometric sequence.

Geometric Sequence

A sequence in which each term after the first is obtained by multiplying the preceding term by a constant is called ageometric sequence.

The constant is denoted by the letter *r* and is called the common ratio. If *a*_{1} is the first term, then the second term is *a*_{1} *r*. The third term is *a*_{1} *r*^{2}, the fourth term is *a*_{1} *r*^{3}, and so on. We can write a formula for the *n*th term of a geometric sequence by following this pattern.

Formula for thenth Term of a Geometric Sequence

Thenth term,a, of a geometric sequence with first term_{n}a_{1}and common ratioris

a=_{n}a_{1}r^{(n-1)}.

**Geometric Sequences**

Definition: Geometric Sequences

A geometric sequence is a sequence in which every number in the sequence is equal to the previous number in the sequence,multipliedby a constant number.

This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example.

**Example — A Flu Epidemic**

Extension: What is influenza?

Influenza (commonly called “the flu”) is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severe illness that most of us get during winter time. The main way that the influenza virus is spread is from person to person in respiratory droplets of coughs and sneezes. (This is called “droplet spread”.) This can happen when droplets from a cough or sneeze of an infected person are propelled (generally, up to a metre) through the air and deposited on the mouth or nose of people nearby. It is good practise to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu.

Assume that you have the flu virus, and you forgot to cover your mouth when two friends came to visit while you were sick in bed. They leave, and the next day they also have the flu. Let’s assume that they in turn spread the virus to two of their friends by the same droplet spread the following day. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner:

Again we can tabulate the events and formulate an equation for the general case:

Figure B: Each person infects two more people with the flu virus.

Day, n |
Number of newly-infected people |
---|---|

1 | 2=2 |

2 | 4=2×2=2×2^{1} |

3 | 8=2×4=2×2×2=2×2^{2} |

4 | 16=2×8=2×2×2×2=2×2^{3} |

5 | 32=2×16=2×2×2×2×2=2×2^{4} |

⋮ | ⋮ |

n |
=2×2×2×2× … ×2=2×2^{(n-1)} |

The above table represents the number of **newly-infected** people after *n* days since you first infected your 2 friends.

You sneeze and the virus is carried over to 2 people who start the chain (*a*_{1}=2). The next day, each one then infects 2 of their friends. Now 4 people are newly-infected. Each of them infects 2 people the third day, and 8 people are infected, and so on. These events can be written as a geometric sequence:

Note the common factor (2) between the events. Recall from the linear arithmetic sequence how the common difference between terms were established. In the geometric sequence we can determine the common ratio,

*r*, by

**Example — A Worm Farmer**

A farmer is breeding worms that he hopes to sell to local shire councils to decompose waste at rubbish dumps. Worms reproduce readily and the farmer expects a 10% increase per week in the mass of worms that he is farming. A 10% increase per week would mean that the mass of worms would increase by a constant factor of (1+10/100) or 1.1.

He starts off with 10 kg of worms. By the beginning of the second week he will expect 10×1.1=11 kg of worms, by the start of the third week he would expect 11×1.1=10×(1.1)

^{2}=12.1 kg of worms, and so on. This is an example of a

**geometric sequence**.

A geometric sequenceis a sequence where each term is obtained by multiplying the preceding term by a certain constant factor.

The first term is 10, and the common factor is 1.10, which represents a 10% increase on the previous term. We can put the results of this example into a table.

From this table we can see that

*t*

_{2}=1.1×

*t*

_{1},

*t*

_{3}=1.1×

*t*

_{2}

and so on. In general:

*t*

_{(n+1)}=1.1×

*t*

_{n}The common factor or common ratio whose value is 1.1 for this example can be found by dividing any two successive terms:

A geometric sequence,

*t*, can be written in terms of the first term,

*a*, and the common ratio,

*r*. Thus:

*a*,

*a*

*r*,

*a*

*r*

^{2},

*a*

*r*

^{3}, …,

*a*

*r*

^{(n-1)}}

The first term

*t*

_{1}=

*a*, the second term

*t*

_{2}=

*a*

*r*, the third term

*t*

_{3}=

*a*

*r*

^{2}, and consequently the

*n*th term,

*t*is

_{n}*a*

*r*

^{(n-1)}.

For a geometric sequence:

t=_{n}ar^{(n-1)}

whereais the first term andrthe common ratio, given by

DEFINITION

Ageometric sequenceis a sequence such that for alln, there is a constantrsuch thata/_{n}a_{(n-1)}=r. The constantris called thecommon ratio.

The recursive definition of a geometric sequence is:

*a*=

_{n}*a*

_{(n-1)}⋅

*r*

When Written in terms of

*a*

_{1}and

*r*, the terms of a geometric sequence are:

*a*

_{1},

*a*

_{2}=

*a*

_{1}

*r*,

*a*

_{3}=

*a*

_{1}

*r*

^{2},

*a*

_{4}=

*a*

_{1}

*r*

^{3}, … .

Each term after the first is obtained by multiplying the previous term by

*r*. Therefore, each term is the product of

*a*

_{1}times

*r*raised to a power that is one less than the number of the term, that is:

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

Since a sequence is a function, we can sketch the function on the coordinate plane. The geometric sequence 1, 1(2), 1(2)

^{2}, 1(2)

^{3}, 1(2)

^{4}or 1, 2, 4, 8, 16 can be written in function notation as {(1, 1), (2, 2), (3, 4), (4, 8), (5, 16)}. Note that since the domain is the set of positive integers, the points on the graph are distinct points that are not connected by a curve.

**Graphs of geometric sequences**

While the graph of an arithmetic sequence is a straight line, the graph of a geometric sequence is a curve for values of *r*>0. (Different Values of *r* produce graphs of different shapes.)

(Warked Example)

Consider the geometric sequence 2, 4, 8, 16, 32, …

(a) Draw up a table showing the term number and its value.

(b) Graph the entries in the table.

(c) Comment on the shape of the graph.

Solution:

(a) Draw up a table showing the term number and its corresponding value.

Term number | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Term value | 2 | 4 | 8 | 16 | 32 |

(b) The value of the term depends on the term number, so ‘Term value’ is graphed on the y-axis. Draw a set of axes with suitable scales. Plot the points and join with a smooth curve.

(c) Comment on the shape of the curve.

The points line on a smooth curve which increases rapidly. Because of this rapid increase in value, it would be difficult to use the graph to predict future values in the sequence with any accuracy.

Shape of graphs of geometric sequences

The shape of the graph of a geometric sequence depends on the value of *r*.

● When *r*>1, the points lie on a curve, as shown in the graph in the previous worked example. Graphs of this kind are said to *diverge* (i.e. they move further and further away from the starting Value).

● When *r*<0, the points *oscillate* on either side of zero.

● When -1<*r*<0, the points *converge* to a certain fixed number, as shown in this graph.

Problem 1. Is the sequence 4, 12, 36, 108, 324, … a geometric sequence?

Solution:

In the sequence,

the ratio of any term to the preceding term is a constant, 3, Therefore, 4, 12, 36, 108, 324, … is a geometric sequence with

*a*

_{1}=4 and

*r*=3.

Problem 2. **Determine whether each sequence is geometric. Write yes or no.**

1). -8, -5, -1, 4, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are not same, the sequence is not geometric.

Answer: No

2). 4, 12, 36, 108, …

Solution:

Find the ratio of the consecutive terms.

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

Answer:

Yes

3). 27, 9, 3, 1, …

Solution:

Find the ratio of the consecutive terms.

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

Answer: Yes

4). 7, 14, 21, 28, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are not the same, the sequence is not geometric.

Answer: No

5). 21, 14, 7, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are not the same. the sequence is not geometric.

Answer: No

6). 124, 186, 248, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are not the same, the sequence is not geometric.

Answer: No

7). -27, 18, -12

Solution:

Find the ratio of the consecutive terms.

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

Answer: Yes

8). 162, 108, 72, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric.

Answer: Yes

9). ½, -¼, 1, -½

Solution:

Find the ratio of the consecutive terms.

-¼÷½=-½

1÷(-¼)=-4

-½÷1=-½

Since the ratios are not same, the sequence is not geometric.

Answer: No

10). -4, -2, 0, 2, …

Solution:

Find the ratio of the consecutive terms.

Since the ratios are not same, the sequence is not geometric.

Answer: No