A geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number.
Notation
This fixed number is called the common ratio, r.
The common ratio can be positive or negative, an integer or a fraction.
The common ratio can be calculated by dividing any term by the one before it.
e.g. Common ratio = t_{ n+1} ÷ t_{ n}
The first term of a geometric sequence is shown by the variable a.
Geometric Sequence

First term, a

Common ratio, r

2, 6, 18, 54, ...

2

6 ÷ 2 = 3 
20, 10, 5, 2.5, ...

20

10 ÷ 20 = 0.5
5 ÷ 10 = 0.5 etc. 
General Term, t_{n}
A geometric sequence can be written:
First term

Second term

Third term

Fourth term

General term (n th term)


t _{1}

t _{2}

t _{3}

t _{4}

...

t_{ n}

a

ar

ar^{2}

ar^{3}

...

ar^{ n − 1}

Example 1

What is the common ratio of the geometric sequence: 20, 10, 5, 2.5,, ... 
Common ratio = t_{ n+1} ÷ t_{ n} Second term ÷ first term = t2 ÷ t1 = 10 ÷ 20 = 0.5 Check: Third term ÷ second term = 5 ÷ 10 = 0.5 The common ratio is 0.5 
Example 2

Find the 8th term of the geometric sequence: 3, 9, 27, 81, ... 
Common ratio, r = 9 ÷ 3 = 3 using t_{n} = ar^{ n − 1} t8 = 3 x 3 ^{8 1} The 8th term is 6561 
Example 3

Which term of the sequence 2, 4, 8, 16, ... would be equal to 1024? 
Common ratio, r = 2 Using t_{n} = ar ^{n1} 1024 =2 x 2 ^{n1} 1024 is the 10th term. 
Example 4

The third term of a geometric sequence is 8 and the fifthterm is 32. Find the first term, a, and the common ratio, r, and thus list the first four terms of the sequence. 
t _{3} = 8 using t_{n} = ar ^{n1} 32 = ar ^{5 − 1} 32 = ar ^{4} 4 = r^{2} 8 = a x 2^{2} The sequence is 2, 4, 8, 16, ... or 2, 4, 8, 16... 