Learning Resources
Overview
Sequences and series are used to describe many situations involving growth, such as money or populations. In Mathematical Modeling, Book 3 you have done some work with sequences; arithmetic and geometric, as well as others. In this unit, you will extend on the knowledge you have gained thus far through opportunities to link your previous knowledge with new concepts.
You will create discrete graphs of sequences. In addition, you will develop formulas for finding the nth term of a sequence and for adding the terms of finite arithmetic and geometric series. You will also explore how to add up the terms of infinite series by considering the concept of a limit. As well, you will investigate various forms of mathematical proofs, both formal and informal, and discover techniques for calculating areas of regions under curves.
This unit has four sections, as outlined below.
Section 1.1: introduces the definition of recursion in the context of sequences. You will explore arithmetic sequences and series both algebraically and visually to see the relationship between recursive and non-recursive formulas.
Section 1.2: focuses on geometric and other recursive sequences. Sequences are graphed and formulas for the general term of a geometric sequence and the sum of a geometric series are developed. You will examine the advantages and disadvantages of using recursive formulas to describe sequences.
Section 1.3: explores the concept of proof by mathematical induction. You will learn about deductive and inductive reasonong, and use the technique of mathematical induction to prove statements involving natural numbers.
Section 1.4: defines the term limit in the context of sequences. You will be introduced to converging and diverging geometric sequences, and to calculation of the area of regions under curves.
This unit of work should take approximately 20 hours to complete.