Math Journal 9-5Evaluate Simplify1) 15 – (-13) = 2)

Find the next 4 terms of the Arithmetic Sequence.3) 7, 4, 1, -2,

Find the next 3 terms of the Recursive Se-quence.4) 1, 3, 4, 7, 11,

Unit 2 Day 4: Sequences as

FunctionsEssential Questions: How can any term of an arithmetic sequence be determined? How do

we represent a sequence in function notation?

Patterns in Arithmetic Sequences

Patterns can be thought of as sequences, or a list of numbers. The below example is what type of sequence?

Example: The set of Natural Numbers

1, 2, 3 , …

Arithmetic

+1+1+1

We can write an arithmetic sequence recursively if we know the pattern (or rule), and the first term. Writing a sequences recursively helps us find any term in the sequence.

Example: 3, 8, 13, 18, … What is the pattern?

+5 is called the common difference.

We can use recursion to find the common difference without ‘guessing’ or ‘analyzing’.

Current term: 8 Previous term: 3

Subtract 3 from the current term:

8 - 3 = +5

First, find the common difference, label it d.

d = 18 – 12 = 6

Now that we determined that this is an arithmetic sequence with a common difference between

successive terms, we can predict the following terms:

Describe the sequence recursively : 12, 18, 24, 30, …

Term # Term

1 12

2 18

3 24

4 30

5

6

36

42

Example 1

Is This Always Useful?

What are some drawbacks?What if we want to find the 100th term in the sequence? We would have to find all 99 terms that precede it!

Arithmetic nth Formula (nth term):

an = a1 + d(n - 1)

Term I want NOW!

1st Term in the Sequence

Common Difference

Term Number

Example 2Use the formula for the following arithmetic sequence,

then find the 10th term:6, 4, 2, 0, …

a1 = 6 n = 10d = 4 - 6 = -2

an = 6 + (-2)(n - 1)a10 = 6 - 2(10 - 1)

a10 = 6 - 2(9)

a10 = 6 - 18

a10 = -12

an = a1 + d(n - 1)

Use the arithmetic formula to determine the 9th term in the sequence: 3, 9, 15, 21, …

an = 3 + 6(9 - 1)a9 = 51

Example 3

Writing An Arithmetic Sequence as a Function

1. List the given sequence.2. Write down the formula:

.3. Identify the first term: 4. Calculate the common difference: 5. Plug and into the Arithmetic nth Formula.

6. Distribute the value.7. Combine all like terms if needed. 8. Change the to function notation a(n).

Consider the sequence 7, 11, 15, 19, … Think of each term as the output of a function. Think of the term number (n) as the input.

Term number (n)

1 2 3 4 input

Term 7 11 15 19 output

Writing An Arithmetic Sequence as a Function

Following the Steps!!1. 7, 11, 15, 19

2. = 73. = (11 - 7) = 44. an = 7 + 4(n - 1)5. an = 7 + 4n – 4 6. an = 4n + 37. a(n) = 4n + 3

Term #

1 2 3 4 input

Term 7 11 15 19 output

Example 4

Write the arithmetic sequence as a function.6, 4, 2, 0, …

an = 6 + -2(n - 1)

an= 6 + -2n + 2an= -2n + 8

a(n) = -2n + 8

Example 5

Write the function for the arithmetic sequence.3, 9, 15, 21, …

an = 3 + 6(n - 1)

an= 3 + 6n - 6an= 6n - 3

a(n) = 6n - 3

Example 6

SummaryEssential Questions: How can any term of an arithmetic sequence be determined? How do we represent a sequence in function notation?

Take 1 minute to write 2 sentences answering the essential questions.