Math 12 – Geometric Sequences

 

A Geometric Sequence is a sequence that has a common ratio.  Just like a common difference can be used to identify an arithmetic sequence, a common ratio is used to identify a geometric sequence.  To find a common difference, you take each number in the sequence and subtract the number before it in the sequence.  If all of the differences are the same then there is a common difference.  To find a common ratio, take each term in the sequence and divide it by the term before it.  If all of the ratios are the same number then there is a common ratio.

 

Example:

{ 2, 4, 6, 8, 10, 12, ... }

D₁ = { (4-2), (6-4), (8-6), (10-8), (12-10), ...}

D₁ = { 2, 2, 2, 2, 2, ...}

Common Difference of 2.  This is an Arithmetic Sequence.

 

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

{ (4÷2), (8÷4), (16÷8), (32÷16), (64÷32) ...}

{ 2, 2, 2, 2, 2, ...}

Common Ratio of 2.  This is a Geometric Sequence.

 

 

The Shape of an Exponential Function

 

The terms of arithmetic sequence form a linear function.  The graph of this linear function will be a straight line.  The terms of a geometric sequence form an exponential function.  The graph of an exponential function will be a curved line that gets steeper and steeper.

 

Exponential Growth  

                                                                                                                        

                        

 

 

 Exponential Decay

 

 

 

 

 

 

The first graph, , shows exponential growth.  Exponential growth can be used to model functions like nuclear reactions, population growth, and investment growth.  The second graph, , shows exponential decay.  Exponential decay can be used to model functions like carbon-14 dating and depreciation of car values.