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 Click here
- how to find the nth term in a geometric sequence Click here
- geometric series Click here
- 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:
- 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
- Given the formula for geometric sequence, determine the first two terms, and then the 5th term. Also state the common ratio.
- 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.
- 3,6,12,24,48,96, …
- 3, 3/2, 3/4, 3/8, 3/16, 3/32, 3/64, …
- 10,15,20,25,30, …
- -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,..
- Derive the an formula.
- Find a10
How to find the general term or nth term of a geometric sequence?
Examples:
- 3, 3/2, 3/4, 3/8, 3/16, …
- 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:
- Determine the sum of the geometric series.
3 + 6 + 12 + … + 1536 - 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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