Topic: Sequences and Series
Subtopic: Geometric Sequences and Series
Overview
Our studies today take us into the second of two special sequences. Sequences such as 5, 10, 20, 40, 80, ... where you multiply a fixed number to get the next terms are called geometric sequences. This particular sequence can be written in general as {5*2n-1}. Again it is important to be able to write the first few terms given this general formula or derive it when given the sequence's terms. Geometric series are sums where the terms are geometric such as 5+10+20+40+80+....
Something to think about: Some infinite geometric series have finite sums. Under what conditions do infinite geometric series have finite sums?
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 7.3.1 Write the first few terms of a geometric sequence given the an term
- 7.3.2 Write the next few terms of a geometric sequence given the first few terms
- 7.3.3 Find a specific term or the total number of terms in a geometric sequence
- 7.3.4 Find a specific term or the total number of terms in a geometric series
- 7.3.5 Write a geometric series in summation notation
- 7.3.6 Use formulae related to geometric sequences and series (both finite and infinite series)
- 7.3.7 Use arithmetic series to convert repeating decimals to fractions
- 7.3.8 Understand a variety of applications involving geometric sequences and series
Terminology
Define: geometric sequence, first term, common ratio, nth term, geometric series, geometric sequence & series formulae
Text Notes
Text:
College Algebra 5ed by Blitzer, sect. 8.3-8.7
- Section 8.3 Concentrate on the algebraic techniques, particularly finding the n-th term, number of terms, and sum.
- Section 8.3 Pay attention to the differences between a finite and infinite geometric sum.
- Section 8.3 Example 9 shows a rather tedious method of converting a repeating decimal to a fraction. It may be somewhat cool, but simple algebra can be used to accomplish the same conversion. If you want to see the steps, ask in class and we can review the elementary algebra method of conversion.
- SKIP sections 8.4-8.7. And that concludes the course, woohoo!