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More Lessons for Intermediate Algebra

More Lessons for Algebra

Math Worksheets

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

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.

a_{n} = a_{n - 1}r

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: 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 a
_{n}formula. - Find a
_{10}

**How to find the general term or nth term of a geometric sequence?**

Examples:

- 3, 3/2, 3/4, 3/8, 3/16, …
- a
_{3}= 5, a_{7}= 80

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.

a_{n}= 2(-3)^{n-1}, n = 5

**How to find the sum of a geometric series?**

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