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Lesson 23: Sequences and series

Topic: This lesson covers Chapter 23: Sequences and series

WeBWorK: There are three WeBWorK assignments on today’s material:Sequences – Introduction

Sequences – Arithmetic

Series – Finite Arithmetic

**Question of the day:** What is a sequence? What is a series?

**Answer of the day: ***A sequence is just a list of numbers. A series is list of numbers, added up.*

#### Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## Introduction to sequences and series

Today we will introduce two very powerful ideas (they are the building blocks of Calculus) – however, the ideas themselves are not complicated: sequences and series. They both have to do with *lists of numbers*, rather than individual numbers.

**Definition 23.1**. A sequence is an ordered list of numbers (we call each number in the list a *term* in the sequence). We write a sequence in order as follows:

In short we write the above sequence as or

Let’s look at examples of some sequences. As you look at this example, keep the following questions in mind:

- Is there a pattern in the sequence?
- Can you predict what the next term of the sequence will be?
- Can you predict what the 10th term of the sequence will be? What about the 20th, or the 100th?
- Can you find a formula for , the th term of the sequence?

**Example 23.2**. Here are some examples of sequences.

a)

b)

c)

d)

e)

**Example 23.3**. Consider the sequence with . Calculate the first five terms of the sequence and . What is the th term of the sequence?

*VIDEO: Introduction to sequences – Examples 23.2 and 23.3*

**Example 23.4**. Find the first 6 terms of each sequence.

a)

b)

c)

d)

Another way to describe a sequence is by giving a **recursive formula** for the th term in terms of the lower terms. Here are some examples.

**Example 23.6**. Find the first terms in the sequence described below.

a) and for

b) and for

c) and for

*VIDEO: Two ways of describing sequences -Examples 23.4 and 23.6*

**Definition 23.8**. A series is a sum of terms in a sequence. We denote the sum of the first terms in a sequence with the following notation:

The summation symbol comes from the greek letter , pronounced “sigma,” which is the greek letter for “S.”

**Example 23.9**. Find the sum.

a) for

b) for

c)

*VIDEO: Intro to series – Example 23.9*

## The arithmetic sequence

**Definition 23.10**. A sequence is called an **arithmetic sequence** if any two consecutive terms have a common difference . The arithmetic sequence is determined by and the first value . This can be written recursively as:

Alternatively, we have the general formula for the th term of the arithmetic sequence

**Example 23.11**. Determine if the sequence is an arithmetic sequence. If so, then find the general formula for .

a)

b)

c)

d)

*VIDEO: Arithmetic sequences – Example 23.11*

**Example 23.12**. Find the general formula of an arithmetic sequence with the given property.

a) and

b) and

c) and

*VIDEO: Arithmetic sequences – finding a general formula, Example 23.12*

**Example 23.13**. Find the sum of the first 100 integers, starting from 1 . In other words, we want to find the sum of .

*VIDEO: Arithmetic series – intro example 23.13*

**Observation 23.14 **. Let be an arithmetic sequence, whose th term is given by the formula . Then, the sum is given by adding precisely times:

**Example 23.15**. Find the value of the arithmetic series.

a) Find the sum for the arithmetic sequence

b) Determine the value of the sum:

c) Find the sum of the first 35 terms of the sequence

*VIDEO: Arithmetic series – finding the sum, Example 23.15*

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