 # Geometric Sequences and Series

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

• geometric sequences
• how to find the nth term in a geometric sequence
• geometric series
• how to find the sum of a geometric series

### Geometric Sequences

A list of numbers that follows a rule is called a sequence. Sequences whose rule is the multiplication of a constant are called geometric sequences, similar to arithmetic sequences that follow a rule of addition. Homework problems on geometric sequences often ask us to find the nth term of a sequence using a formula. Geometric sequences are important to understanding geometric series.

Geometric Sequences
Determine the nth term of a geometric sequence.
Determine the common ratio of a geometric sequence.
Determine the formula for a geometric sequence.

A geometric sequence is a sequence that has the pattern of multiplying by a constant to determine the consecutive terms.
We say geometric sequences have a common ratio.
an = an - 1r
Example:
1. A sequence is a function. What is the domain and range of the following sequence? What is r?
-12, 6, -3, 3/2, -3/4

2. Given the formula for geometric sequence, determine the first two terms, and then the 5th term. Also state the common ratio.

3. Given the geometric sequence, determine the formula, Then determine the 6th term.
1/3, 2/8, 4/17, 8/81, ...
Geometric Sequences (Introduction)

A Quick Intro to Geometric Sequences
Gives the definition of a geometric sequence and go through 4 examples, determining if each qualifies as a geometric sequence or not.
A geometric sequence is a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed non-zero number, called the common ratio.
Examples:
Determine which of the following sequences are geometric. If so, give the value of the common ratio, r.
1. 3,6,12,24,48,96, ...
2. 3, 3/2, 3/4, 3/8, 3/16, 3/32, 3/64, ...
3. 10,15,20,25,30, ...
4. -1, .1, -.01, .001, -.0001

### Geometric Sequences - Find the nth term

Geometric Sequences: A Formula for the' n - th ' Term.
Derive the formula to find the 'n-th' term of a geometric sequence by considering an example.
The formula to find another term of the sequence.
Example:
Consider the geometric sequence
3,6,12,24,48,..
1. Derive the an formula.
2. Find a10
How to find the general term or nth term of a geometric sequence?
Examples:
1. 3, 3/2, 3/4, 3/8, 3/16, ...
2. a3 = 5, a7 = 80

### Geometric Series

We can use what we know of geometric sequences to understand geometric series. A geometric series is a series or summation that sums the terms of a geometric sequence. There are methods and formulas we can use to find the value of a geometric series. It can be helpful for understanding geometric series to understand arithmetic series, and both concepts will be used in upper-level Calculus topics.
How to determine the partial sum of a geometric series? Summing or adding the terms of a geometric sequence creates what is called a series.
Example:
1. Determine the sum of the geometric series.
3 + 6 + 12 + ... + 1536
2. Determine the sum of the geometric series.
an = 2(-3)n-1, n = 5

How to find the sum of a geometric series?

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