# Chapter 7: Sequences, Series, and Mathematical Induction

## Introduction

Sequences and series are really just patterns of numbers, and numerical patterns are everywhere. Patterns are found in the growth of trees, the enrollment in an online group, the movement of a bird, the layout of atoms in a crystal, and a million other places.

Understanding how numerical patterns work is a fundamental step toward predicting future events. Imagine using numbers to predict the best way to throw a baseball, or pick the best pitcher out of a collection of free agents. Number series appear in the way people choose to buy products, or donate money.

In this chapter, you will explore the topics above and will learn to both use formulas designed to help you create series, and create formulas *from* series in order to predict later members of sequences.

## Chapter Outline

- 7.1. Recursive Formulas
- 7.2. Explicit Formulas
- 7.3. Sum Notation and Properties of Sigma
- 7.4. Series Sums and Gauss' Formula
- 7.5. Problem Solving with Series Sums
- 7.6. Inductive Proofs
- 7.7. Induction and Factors
- 7.8. Induction and Inequalities
- 7.9. Sums of Finite Geometric Series
- 7.10. Sums of Infinite Geometric Series
- 7.11. Factorials and Combinations
- 7.12. Binomial Theorem and Expansions

### Chapter Summary

## Summary

This chapter covers the application and creation of arithmetic and geometric formulas. Students are introduced to Sigma (sum) notation and are taught to calculate the sum of a partial or infinite series.

Later in the chapter, students will learn about induction and inductive proofs. The chapter wraps up with combinations/permutations and factorials and an introduction to the binomial theorem and binomial expansion.