Section C: Sequences and Series

Unit 4: Algebra II – Advanced Concepts

Section C: Sequences and Series

Welcome

Welcome to Section C: Sequences and Series! In this section, you’ll explore the fascinating world of sequences and series, including arithmetic and geometric sequences, special series, and their applications in various fields.

Imagine

Imagine you’re a financial analyst calculating the future value of an investment. Understanding sequences and series allows you to model the growth of the investment over time, making informed financial decisions.

Context

Previously, you’ve studied basic algebraic functions and their properties. Now, we’ll extend those ideas to sequences and series, where you’ll learn to analyze and sum these mathematical structures, gaining deeper insights into their applications.

Overview

This section covers arithmetic and geometric sequences, finding sums of series, special series like the Fibonacci sequence, using sigma notation, and applying sequences and series in computing and finance. You’ll learn to apply these concepts to real-world problems and enhance your problem-solving skills.

Objectives

  • Understand and analyze arithmetic and geometric sequences, including their properties and applications.
  • Calculate the sum of a series, understanding the techniques used for summing finite and infinite series.
  • Explore special series, such as the Fibonacci sequence, and understand their significance and applications.
  • Use sigma notation to represent and analyze series, gaining proficiency in summation techniques.
  • Apply sequences and series to real-world problems, including computing and finance, understanding their practical applications.

Preparatory Guidance

Definitions and Pronunciations
  • Sequence: An ordered list of numbers, each of which is called a term.
  • Series: The sum of the terms of a sequence.
  • Arithmetic Sequence: A sequence in which the difference between consecutive terms is constant, denoted as d.
  • Geometric Sequence: A sequence in which each term is obtained by multiplying the previous term by a constant, denoted as r.
  • Sigma Notation: A concise way of writing the sum of a series, using the Greek letter \Sigma.
Verbal Reading of Equations
  • For a_n = a_1 + (n-1)d, say “a n equals a one plus n minus one times d.”
  • For S_n = \frac{n}{2}(a_1 + a_n), say “S n equals n over two times a one plus a n.”
  • For S = \frac{a_1}{1 - r}, say “S equals a one over one minus r.”
Problem-Solving Strategies
  • Analyze arithmetic and geometric sequences, understanding their properties and applications.
  • Calculate the sum of a series, understanding the techniques used for summing finite and infinite series.
  • Explore special series, such as the Fibonacci sequence, and understand their significance and applications.
  • Use sigma notation to represent and analyze series, gaining proficiency in summation techniques.
  • Apply sequences and series to real-world problems, including computing and finance, understanding their practical applications.

Considerations

How do sequences and series apply to real-world problems? Why is it important to understand and analyze these mathematical structures in various fields? In what ways can you use these concepts in your daily life or future career?