Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: College Algebra
Topic: Sequences and Series
Subtopic: Introduction to Sequences and Series

Overview

Pop quiz! Can you complete these sequences?
a. 5, 11, 17, 23, 29, ___, ___, ___
b. 2, 4, 8, 16, 32, ___, ___, ___
c. 3, 5, 7, 11, 13, ___, ___, ___
d. 1, 1, 2, 3, 5, 8, ___, ___, ___

a. 35, 41, 47. To get the subsequent terms you add 6. This is an example of an arithmetic sequence.
b. 64, 128, 256. To get the subsequent terms you multiply by 2. This is an example of a geometric sequence.
c. 17, 19, 23. This is simply a sequence of prime numbers.
d. 13, 21, 34. Can you see that the subsequent terms are formed by combining previous terms (1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.)? This is an example of a recursive sequence.

This lesson introduces general sequences (lists) and series (sums) -- the notation, terminology, and processes. We'll follow this lesson with a concentration on specific types of sequences and series (arithmetic and geometric). This material will be used particularly in Calculus III when we will write functions such as ex as a sum of infinitely many rational terms (e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ...). This will enable us to perform calculus on rational expressions rather than on the (more complicated) transcendental function itself.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 8.1.1 Evaluate or simplify a factorial algebraically and electronically
• 8.1.2 Write the first few terms of a sequence given the an term
• 8.1.3 Write the next few terms of a sequence given the first few terms
• 8.1.4 Understand alternating sequences and recursive sequences
• 8.1.5 Know the difference between sequences and series
• 8.1.6 Evaluate or simplify a series algebraically and electronically
• 8.1.7 Use proper mathematical notation and format when working with factorials, sequences, and series
• 8.1.8 Apply properties of series as needed

Terminology

Terms you should be able to define: sequence, finite sequence, infinite sequence, series, finite series, infinite series, terms (of a sequence or series), summation, summation notation, sigma notation, upper limit, lower limit, index of summation, sum, partial sum, sequence of partial sums, factorial, alternating sequence, Fibonacci sequence, recursively-defined (a.k.a. recursive) sequence, properties of series

Text Notes

The Fibonacci Sequence is the most famous example of a recursive sequence. If you haven't investigated this very rich topic before or would like to learn more, please explore the links below.

If your text discusses electronically producing "dot" graphs of a sequence you may SKIP these examples/problems throughout the chapter.

Supplemental Resources (optional)

Pages to explore the Fibonacci sequence and its connections to Pascal's Triangle and the Golden Mean:
Biography of Leonardo Fibonacci
Dr. Knott's Fibonacci Numbers and Nature ~ a must read!
Pascal's Triangle and its Patterns
The Golden Ratio/Section/Mean/Number
Want more sites to explore or some books to read? Just ask! This is a favourite topic of mine, practically a hobby :)