9.3 - Geometric Sequences and Series

Definition:  

A Sequence in which the ratios of consecutive terms are the same
a2/a1=a3/a2=....= r
the number r is the common ratio of the sequence.
Example: 2,4,8,16,32,64,....

The nth Term of a Geometric Sequence:

Expressed in the form
Where r is the common ratio of consecutive terms of the sequence.
Can be written as

Example: Find the 9th term of the sequence 5,15,45

r =15/3=3  so therefore the equation  can be used

Plug in n and r to get

Then simplify to get a9 =32,805

Sum of a Finite Geometric Sequence:

Definition: All of the terms in a series up to and including the nth term added together

finite means that only the terms up until a certain point are being added, not all of the terms

The formula is used

Sum of an Infinite Geometric Series:

Definition: All of the terms in a series from the first term until infinity added together.

|r| must be less than 1 or the sum will be infinity as well.

The sum of an infinite number of terms is only quantifiable because when n approaches infinity, the value of the nth term approaches 0.

Therefore the equation can be expressed without n as a variable.

So is the equation used.

Example:

So the terms of the sequence are

10,1,0.1,0.01,0.001,0.0001,....

Therefore the sum of all of these terms to infinity is 11.1111111....