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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: $a_{1}, a_{2}, a_{3}, a_{4},…$

In short we write the above sequence as $\left\{a_{n}\right\}$ or $\left\{a_{n}\right\}_{n \geq 1}$

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 $a_n$, the $n$th term of the sequence?

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

a) $4,6,8,10,12,14,16,18, \dots$

b) $1,3,9,27,81,243, \dots$

c) $+5,-5,+5,-5,+5,-5, \dots$

d) $1,1,2,3,5,8,13,21,34,55, \dots$

e) $5,8,-12,4,5.3,7,-3, \sqrt{2}, 18, \frac{2}{3}, 9, \dots$

**Example 23.3**. Consider the sequence $\left\{a_{n}\right\}$ with $a_{n}=4 n+3$. Calculate the first five terms of the sequence $a_1, a_2, a_3, a_4,$ and $a_5$. What is the $200$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) $a_{n}=n^{2}$

b) $a_{n}=\frac{n}{n+1}$

c) $a_{n}=(-1)^{n}$

d) $a_{n}=(-1)^{n+1} \cdot 2^{n}$

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

**Example 23.6**. Find the first $6$ terms in the sequence described below.

a) $a_{1}=4,$ and $a_{n}=a_{n-1}+5$ for $n>1$

b) $a_{1}=3,$ and $a_{n}=2 \cdot a_{n-1}$ for $n>1$

c) $a_{1}=1, a_{2}=1,$ and $a_{n}=a_{n-1}+a_{n-2}$ for $n>2$

*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 $k$ terms in a sequence with the following notation:

$\sum_{i=1}^{k} a_{i}=a_{1}+a_{2}+\cdots+a_{k}$

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

**Example 23.9**. Find the sum.

a) $\sum_{i=1}^{4} a_{i},$ for $a_{i}=7 i+3$

b) $\sum_{j=1}^{6} a_{j},$ for $a_{n}=(-2)^{n}$

c) $\sum_{k=1}^{5}\left(4+k^{2}\right)$

*VIDEO: Intro to series – Example 23.9*

## The arithmetic sequence

**Definition 23.10**. A sequence $\left{a_{n}\right}$ is called an **arithmetic sequence** if any two consecutive terms have a common difference $d$. The arithmetic sequence is determined by $d$ and the first value $a_{1}$. This can be written recursively as:

$a_{n}=a_{n-1}+d \quad \text { for } n \geq 2$

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

$a_{n}=a_{1}+d \cdot(n-1)$

**Example 23.11**. Determine if the sequence is an arithmetic sequence. If so, then find the general formula for $a_{n}$.

a) $7,13,19,25,31, \ldots$

b) $13,9,5,1,-3,-7, \dots$

c) $10,13,16,20,23, \dots$

d) $a_{n}=8 \cdot n+3$

*VIDEO: Arithmetic sequences – Example 23.11*

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

a) $d=12,$ and $a_{6}=68$

b) $a_{1}=-5,$ and $a_{9}=27$

c) $a_{5}=38,$ and $a_{16}=115$

*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 $1+2+3+\cdots+99+100$.

*VIDEO: Arithmetic series – intro example 23.13*

**Observation 23.14 **. Let $\left\{a_{n}\right\}$ be an arithmetic sequence, whose $n$ th term is given by the formula $a_{n}=a_{1}+d(n-1)$. Then, the sum $a_{1}+a_{2}+\cdots+a_{k-1}+a_{k}$ is given by adding $\left(a_{1}+a_{k}\right)$ precisely $\frac{k}{2}$ times:

$\sum_{i=1}^{k} a_{i}=\frac{k}{2} \cdot\left(a_{1}+a_{k}\right)$

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

a) Find the sum $a_{1}+\cdots+a_{60}$ for the arithmetic sequence $a_{n}=2+13(n-1)$

b) Determine the value of the sum: $\quad \sum_{j=1}^{1001}(5-6 j)$

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

$4,3.5,3,2.5,2,1.5, \ldots$

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