Download Unit 2 Day 4: Sequences as Functions

Evaluate
Math Journal 9-5
1) 15 – (-13) =
Simplify
2) 14𝑝2 − 3𝑝2 + 𝑝2 − 7𝑝
Find the next 4 terms of the Arithmetic Sequence.
3) 7, 4, 1, -2,
Find the next 3 terms of the Recursive Sequence.
4) 1, 3, 4, 7, 11,
Unit 2 Day 4:
Sequences as
Functions
Essential 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?
Arithmetic
1, 2 , 3 , …
+1 +1 +1
Example: The set of Natural Numbers
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?
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
+5 is called the common difference.
Example 1
Describe the sequence recursively : 12, 18, 24, 30, …
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:
Term #
Term
1
12
2
18
3
24
4
5
30
36
6
42
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 2
an = a1 + d(n - 1)
Use the formula for the following arithmetic sequence, then find
the 10th term:
6, 4, 2, 0, …
d = 4 - 6 = -2
a1 = 6 n = 10
an = 6 + (-2)(n - 1)
a10 = 6 - 2(10 - 1)
a10 = 6 - 2(9)
a10 = 6 - 18
a10 = -12
Example 3
Use the arithmetic formula to determine the
9th term in the sequence: 3, 9, 15, 21, …
𝑎𝑛 = 𝑎1 + 𝑑 𝑛 − 1
𝑎1 = 3, 𝑑 = 6, 𝑛 = 9
an = 3 + 6(9 - 1)
a9 = 51
Writing An Arithmetic Sequence as a
Function
1. List the given sequence.
2. Write down the formula:
𝑎𝑛 = 𝑎1 + 𝑑 𝑛 − 1
.3. Identify the first term: 𝑎1 =
4. Calculate the common difference: 𝑑 =
5. Plug 𝑎1 and 𝑑 into the Arithmetic nth Formula.
6. Distribute the 𝑑 value.
7. Combine all like terms if needed.
8. Change the 𝑎𝑛 to function notation a(n).
Writing An Arithmetic Sequence as a
Function
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
Example 4
Following the Steps!!
1. 7, 11, 15, 19
2. 𝑎𝑛 = 𝑎1 + 𝑑 𝑛 − 1
3. 𝑎1 = 7
4. 𝑑 = (11 - 7) = 4
5. an = 7 + 4(n - 1)
6. an = 7 + 4n – 4
7. an = 4n + 3
8. a(n) = 4n + 3
Term # 1
2
Term
11 15 19 output
7
3
4
input
Example 5
Write the arithmetic sequence as a
function.
6, 4, 2, 0, …
𝒂𝒏 = 𝒂𝟏 + 𝒅(𝒏 − 𝟏)
an = 6 + -2(n - 1)
an= 6 + -2n + 2
an= -2n + 8
a(n) = -2n + 8
Example 6
Write the function for the arithmetic
sequence.
3, 9, 15, 21, … 𝒂𝒏 = 𝒂𝟏 + 𝒅(𝒏 − 𝟏)
an = 3 + 6(n - 1)
an= 3 + 6n - 6
an= 6n - 3
a(n) = 6n - 3
Summary
Essential 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.