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*2^n-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:

- 8.3.1 Write the first few terms of a geometric sequence given the a
_{n}term - 8.3.2 Write the next few terms of a geometric sequence given the first few terms
- 8.3.3 Find a specific term or the total number of terms in a geometric sequence
- 8.3.4 Find a specific term or the total number of terms in a geometric series
- 8.3.5 Write a geometric series in summation notation
- 8.3.6 Use formulae related to geometric sequences and series (both finite and infinite series)
- 8.3.7 Use arithmetic series to convert repeating decimals to fractions
- 8.3.8 Understand a variety of applications involving geometric sequences and series

**Terminology**

Terms you should be able to define: geometric sequence, first term, common ratio, n^{th} term, geometric series, geometric sequence & series formulae

**Text Notes**

Concentrate on the algebraic techniques, particularly finding the n-th term, number of terms, and sum. Pay attention to the differences between a finite and infinite geometric sum.