Geometric sequence

A geometric sequence is a sequence where each term is found by multiplying or dividing the same value from one term to the next. This value that we multiply or divide is called "common ratio" 

A sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.

For example, the following two sequences are examples of geometric sequences.

2, 4, 8, 16, 32, 64, ...

243, 81, 27, 9, 3, 1, ...

Geometric sequence

Looking carefully at 2, 4, 8, 16, 32, 64, ......., helps us to make the following observation:

As you can see, each term is found by multiplying 2, a common ratio, to the previous term. Notice that we have to add 2 to the first term to get the second term, but we have to add 4 to the second term to get 8. This shows indeed that this sequence is not created by adding or subtracting a common term.

Geometric sequence

Looking at 243, 81, 27, 9, 3, 1, ... carefully helps us to make the following observation:

This time, to find each term, we divide by 3, a common ratio, from the previous term.

How to model a geometric sequence with an exponential function

Many geometric sequences can me modeled with an exponential function. An exponential function is a function of the form an where a ≠ 1 and n is a variable.

Here is a trick or "recipe per se" to quickly get an exponential function!

1) Let us try to model 2, 4, 8, 16, 32, 64, .......

Let n represent any term number in the sequence.

Observe that the terms of the sequence can be written as 21, 22, 23, ......

We can therefore model the sequence with this exponential function: 2n

Check to see if the exponential function works:

  • When n = 1, which represents the first term, we get 21 = 2
  • When n = 2, which represents the second term, we get 22 = 4
  • When n = 3, which represents the third term, we get 23 = 8

The exponential function works!


2)
 Let us try to model 243, 81, 27, 9, 3, 1, .......

Let n represent any term number in the sequence.

Observe that the terms of the sequence can be written as 35, 34, 33, .......

We just have to model the sequence: 5, 4, 3, .....

The process will be briefly explained here. For detailed explanation, see arithmetic sequence.

The number we subtract to each term is 1.

The number that comes right before 5 in the sequence is 6.

5, 4, 3, ..... can be modeled with the following algebraic expression: -1 × n + 6

We can therefore model 243, 81, 27, 9, 3, 1, ............... with the exponential function below: 3-n + 6

Check to see if the exponential function works:

  • When n = 1, which represents the first term, we get 3-1 + 6 = 35 = 243
  • When n = 2, which represents the second term, we get 3-2 + 6 = 34 = 81
  • When n = 3, which represents the third term, we get 3-3 + 6 = 33 = 27

The exponential function works!

Geometric sequence formula

1)

2n is the exponential function for 2, 4, 8, 16, 32, 64, .......

Let us try to rewrite 2n by making the first term appear in the exponential function.

2n = 2 × 2n × 2-1 (since 2 × 2-1 = 20 = 1 and 1 × 2n = 2n)

2n = 2 × 2n - 1 

2 is the number we multiply to each term

2 is the first term

n is the number of terms

2)

3-n + 6 is the exponential function for 243, 81, 27, 9, 3, 1, ............

Let us try to rewrite 3-n + 6 by making the first term, also equal to 35, appear in the exponential function.

3-n + 6 = 35 × 3-n + 6 × 3-5 (since 35 × 3-5 = 30 = 1 and 1 × 3-n + 6 = 3-n + 6)

3-n + 6 = 35 × 3-n + 1

3-n + 6 = 35 × 3-1(n - 1)

3-n + 6 = 35 × (1 / 3)n - 1 (3-1 = 1 / 3)

1 / 3 is the number we multiply to each term and it means the same as dividing each term by 3.

35 is the first term

n is the number of terms

In general, 

Let r be the number we multiply each time or the common ratio.

Let a1 be the first term

Let n be the number of terms

Let an be the nth term.

Then, an = a1 × rn - 1 

A couple of exercises about geometric sequences

Are the given sequences geometric? If so, find the 9th term.

a. 20, 5, 1, 0.25, ....

b. 12, 3, 0.75, 0.1875, ....

20, 5, 1, 0.25, .... is not a geometric sequence since the number we divide each term by is not always the same.

12, 3, 0.75, 0.1875, .... is a geometric sequence since the number we always divide each term by is 4.

Then, an = a1 × rn - 1 

r = 1 / 4 = 0.25

n = 9

a1 = 12

a9 = 12 × (0.25)9-1

a9 = 12 × (0.25)8

a9 = 12 × 0.00001525878

a9 = 0.00018310546

Take the geometric sequence quiz below to check your understanding of this lesson.

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