Geometric sequences -

Definition 1: Geometric sequence: A geometric sequence is a sequence of numbers in which each new term (except for the first term) is calculated by multiplying the previous term by a constant value called the constant ratio (r).

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

Example: A flu epidemic

Influenza (commonly called “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 influenza virus is spread 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 through the air and deposited on the mouth or nose of people nearby. It is good practice to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu. Regular hand washing is an effective way to prevent the spread of infection and illness.

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 each friend in turn spreads 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:

Each person infects two more people with the flu virus.

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

Table 1
Day (n)	No. of newly-infected people
1	2=2
2	4=2×2=2×21
3	8=2×4=2×2×2=2×22
4	16=2×8=2×2×2×2=2×23
5	32=2×16=2×2×2×2×2=2×24
⋮	⋮
n	2×2×2×2×⋯×2=2×2n−1

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

You sneeze and the virus is carried over to 2 people who start the chain (a=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 new people are infected, and so on. These events can be written as a geometric sequence:

2; 4; 8; 16; 32; \dots

Note the constant ratio (r=2) between the events. Recall from the linear arithmetic sequence how the common difference between terms was established. In the geometric sequence we can determine the constant ratio (r) from:

T 2 T 1 = T 3 T 2 = r

More generally,

T n T n - 1 = r

Exercise 1: Constant ratio of a geometric sequence

Problem 1:

Determine the constant ratios for the following geometric sequences and write down the next three terms in each sequence:

5;10;20;…
12;14;18;…
7;0,7;0,07;…
p;3p2;9p3;…
−3;30;−300;…

The general term for a geometric sequence

From the flu example above we know that T1=2 and r=2, and we have seen from the table that the n term is given by Tn=2×2n−1.

The general geometric sequence can be expressed as:

T 1 T 2 T 3 T 4 T n = a = a \times r = a \times r \times r = a \times r \times r \times r = a \times [r \times r \dots (n - 1) times] = a r 0 = a r 1 = a r 2 = a r 3 = a r n - 1

Identity 1

Therefore the general formula for a geometric sequence is:

T n = a r n - 1

where

a is the first term in the sequence;
r is the constant ratio.

Test for a geometric sequence

To test whether a sequence is a geometric sequence or not, check if the ratio between any two consecutive terms is constant:

T 2 T 1 = T 3 T 2 = T n T n - 1 = r

If this condition does not hold, then the sequence is not a geometric sequence.

Exercise 2: General term of a geometric sequence

Problem 1:

Determine the general formula for the n term of each of the following geometric sequences:

5;10;20;…
12;14;18;…
7;0,7;0,07;…
p;3p2;9p3;…
−3;30;−300;…

Example 1: Flu epidemic

Question

We continue with the previous flu example, where Tn is the number of newly-infected people after n days:

T n = 2 \times 2 n - 1

Calculate how many newly-infected people there are on the 10 day.
On which day will 16 384 people be newly-infected?

For this geometric sequence, plotting the number of newly-infected people (Tn) vs. the number of days (n) results in the following graph:

Table 2
Day (n)	No. of newly-infected people
1	2
2	4
3	8
4	16
5	32
6	64
n	2×2n−1

In this example we are only dealing with positive integers (n∈{1;2;3;…},Tn∈{1;2;3;…}), therefore the graph is not continuous and we do not join the points with a curve (the dotted line has been drawn to indicate the shape of an exponential graph).

Geometric mean

The geometric mean between two numbers is the value that forms a geometric sequence together with the two numbers.

For example, the geometric mean between 5 and 20 is the number that has to be inserted between 5 and 20 to form the geometric sequence: 5;x;20

Determine the constant ratio: x 5 ∴ x 2 x 2 x = 20 x = 20 \times 5 = 100 = \pm 10

Important: remember to include both the positive and negative square root. The geometric mean generates two possible geometric sequences:

5; 10; 20; \dots

5; - 10; 20; \dots

In general, the geometric mean (x) between two numbers a and b forms a geometric sequence with a and b:

For a geometric sequence: a; x; b

Determine the constant ratio: x a x 2 ∴ x = b x = a b = \pm a b --\sqrt

Exercise 3: Mixed exercises

Problem 1:

The n term of a sequence is given by the formula Tn=6(13)n−1.

Write down the first three terms of the sequence.
What type of sequence is this?

Problem 2:

Consider the following terms:

(k - 4); (k + 1); m; 5 k

The first three terms form an arithmetic sequence and the last three terms form a geometric sequence. Determine the values of k and m if both are positive integers.

[IEB, Nov 2006]

Problem 3:

Given a geometric sequence with second term 12 and ninth term 64.