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" 

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

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

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

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 × 2^{n - 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

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, a_n = a₁ × r^{n - 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.