WARM UP/GO OVER IT: 15 MINUTES ANNOUNCEMENTS/NEW UNIT: 1 MINUTE PATTERNS/MAKE YOUR OWN: 10 MINUTES EXPLICIT AND RECURSIVE: 14 MINUTES ARITHMETIC SEQUENCES: 40 MINUTES CLASS WORK: 10 MINUTES AFM Unit 5 – Sequences and Series

Warm Up 1. A catering company’s charges are given by the table below: a. Write a piecewise function to represent the charge rules in the table above. b. Graph your piecewise function (pick appropriate scales for your graph). c. Is the function continuous? d. How much would it cost to cater a wedding with 170 people? e. If my budget is $3000, can I throw an event for 300 people? Number of peopleCost Up to 25 people$500 Between 25 and 400 people$500 plus $15 per person over the first or more people$5625 plus $10 per person over the first 400

Number of peopleCost Up to 25 people$500 Between 25 and 400 people$500 plus $15 per person over the first or more people$5625 plus $10 per person over the first 400

Number of peopleCost Up to 25 people$500 Between 25 and 400 people$500 plus $15 per person over the first or more people$5625 plus $10 per person over the first 400

How much would it cost to cater a wedding with 170 people? If I have a budget of $3000, can I throw an event for 300 people? Number of peopleCost Up to 25 people$500 Between 25 and 400 people$500 plus $15 per person over the first or more people$5625 plus $10 per person over the first 400

Announcements You have until TOMORROW (Friday) afternoon to complete any and all make up work for this quarter. This includes finishing your midterm and make up tests. I will be here today and tomorrow after school.

Unit 5 – Sequences and Series Our goals for this unit:  Know what arithmetic and geometric sequences are.  Translate between recursive and explicit representations of them  Determine whether a sequences converges or diverges  Find the sum of finite and infinite arithmetic and geometric series.

Sequences Lists of numbers that follow a pattern. Terms: the numbers in the sequence. Term number: where the number falls in the sequence Example: 2, 4, 6, 8, 10… a 1 a 2 a 3 a 4 a 5

What is the next number in the pattern? 7, 14, 21, 28, 35, ___

What is the next number in the pattern? 0, 1, 3, 6, 10, 15, ___

What is the next number in the pattern? 1, 2, 4, 8, 16, 32, 64, 128, ___

What is the next number in the pattern? 3, 1, 2, 0, 1, -1, ___

What is the next number in the pattern? 1, 8, 27, 64, 125, 216, ___

What is the next number in the pattern? 1, -1, 2, 0, 3, ___

What is the next number in the pattern? 1, 1, 2, 3, 5, 8, 13, 21, ____

Stump your neighbor Your goal: stump your neighbor with your brilliant pattern Create your own pattern/rule, and write down the first four terms in your sequence You have 2 minutes to come up with your rule and first 4 terms Then, you will switch with a neighbor and try to figure each other’s pattern out.

Writing Rules for Patterns We can write pattern rules using equations. This helps us because we can use that pattern rule to find any term in the sequence that we want, from the first term to the 100 th term, or even higher.

Example Example: Find the first four terms of the sequence given by the rule: a n = 2n(-1) n First term: a 1 Second term: Third term: Fourth term:

You try the next two. Write the first 6 terms of the sequence a n = n 3 – 10 Write the first 5 terms of the sequence a n = -2 n + 7

Explicit Formula The type of formula we just looked at is called an explicit formula. We plugged in our term number “n”, and it told us the value of a term. There is also another common way to write a rule for a pattern.

Recursive Formula With a recursive formula, you are given: the first term of the sequence What you have to do to the previous term to get the one you want.

Recursive Formula in Words Example: The first term of a sequence is 5. To get the next term of the sequence, you add 3 to the term before it. What are the first 3 terms of the sequence? What is the 6 th term of the sequence?

Recursive Formula Example: Find the fifth term of the recursively defined sequence a 1 = 1, a n = a n – 1 + 2n – 1 Notation to know:  a n – 1 represents the term immediately before a n  a n – 2 represents the term two terms before a n

You try two. Find the first 5 terms of the sequence given by: a 1 = 3, a n = (-2)a n – 1 Find the 6 th term of the sequence given by: a 1 = 8, a n = 2a n – 1 – 7

Types of Sequences While there are many different types of sequences, we are going to focus on two in this class: Arithmetic Sequences Geometric Sequences

Arithmetic Sequences The difference between successive terms is constant. In other words…the terms go up or down by the same amount every single time. Examples: 3, 7, 11, 15, 19… or 10, 8, 6, 4, 2, 0, -2…

Are the following sequences arithmetic?

Common difference The amount by which the terms increase or decrease each time. We use the variable “d” to represent this. We find it by subtracting. Example: What is the common difference of the terms in the sequence below? 5, 8, 11, 14, 17, 20…

Determine the common difference and use it to find the next four terms of the sequences below. 17, 12, 7, … d = Next 4 terms: 3, 16, 29, … d = Next 4 terms:

How did you use the common difference to find the next terms in the sequences? You figured out what it was, and then added that to the previous term to get the next one. Basically, you wrote a recursive formula in your head! a 1 = __, a n = a n-1 + d So let’s go back and write out the recursive formulas for the last sequences

Write the recursive formula for the sequences below. 17, 12, 7, … d = Recursive Formula: 3, 16, 29, … d = Recursive Formula:

You try two. Write the recursive formulas for the sequences below. 117, 108, 99, … -3, 1, 5, …

Recursive vs. Explicit Given the following recursive formula: a 1 = 3, a n = 2a n – a. Find the 5 th term b. Find the 10 th term c. Find the 100 th term It’s a huge pain to find the 100 th term with a recursive sequence. Is there a more efficient way?

Why, yes, there is! Remember the explicit formula from earlier? All we had to do was plug in the term number “n” into the formula and it gave us the term we were looking for. So how do we write an explicit formula?

Let’s look at our recursive formula for a minute. Let’s use the recursive formula where a 1 = 6 and d = 3 We can write a 1 = 6 and a n = a n – First terma1a1 a1a1 6 Second terma2a2 a 1 + d6 + 3 = 9 Third term Fourth term Fifth term Nth term

So our explicit formula is… a n = a 1 + (n – 1)d

Examples: Find both an explicit and recursive formula for the nth term of the arithmetic sequences below: 35, 23, 11, … 2, 5, 8, …

You Try two: 4, 19, 34, … 25, 11, -3, …

We can use these formulas to find any missing value we want. d (common difference) a n (nth term) a 1 (first term) n (term number)

Example: Find the 68 th term of the arithmetic sequence 25, 17, 9, …

Example: Find the first term of the arithmetic sequence where a 25 = 139 and d = ¾

Find the common difference of the arithmetic sequence for which a 1 = 75 and a 38 = 56.5

Practice Time Complete the problems on the worksheet I give you. You may work with your neighbors. Whatever you do not finish is homework!

Homework Finish class work!