1. Arithmetic Sequences as functions
Unit 2 Lesson 3
Arithmetic Sequences as functions

2. What is an arithmetic sequence?
A list of numbers that adds or subtracts the same amount to get to the next term (terms are positions in the sequence and are separated by commas)
Example:
1, 4, 7, 10, 13, 16, ….
Notice in this sequence we are increasing by 3 to get to the next term.

3. Writing a sequence as a function rule…
Let’s begin by listing our sequence as we would a function with input and output values. Using our previous example: 1, 4, 7, 10, 13, 16,… we can create the following table of values.
Input
(term)
Output (value of the term) 1 2 4 3 7 10 5 13 6 16

4. Continued…. Now let’s graph our table on a coordinate plane.
Output (value of the term)
Notice how the graph appears to be linear.
Input (term)

5. Continued….
What if we used what we know about linear functions to write a function rule for this sequence? Let’s use point slope form to write the equation of our line. y – y1 = m(x – x1) Solving for y gives us: y = m(x – x1) + y1

6. Continued….. Input (term) Output (value of the term) 1 2 4 3 7 10 5 136 16
First we need to find the slope or rate of change.
Using two points (1, 1) and (2, 4) we can find the slope.
m =
Next we need to use the point (1, 1) (our first input and output) to substitute into our formula…
y = m(x – x1 ) + y1
y = 3(x – 1) + 1

7. Final steps…
Finally we have the equation for our function rule. y = 3(x – 1) + 1 We can simplify the equation by distributing and combining like terms…. y = 3x – 2 Now how do we turn this into sequence notation?

8. Arithmetic Sequence Definition
The formula for writing an explicit definition for an arithmetic sequence is: An = d(n – 1) + A1.
In this formula:
An is representing the values of the terms and is acting like y from our earlier function.
d is called the constant difference and is acting like the slope, m
n is representing the term and is acting like x
(1, A1) is the (1st term, value of the 1st term) and is acting like the point we are substituting.

9. Continued…
Now we can use the formula An = d(n-1) + A1 to write the definition of our sequence. d = 3 because each term is increasing by three each time. A1 = 1 because the value of the first term is 1. Substituting into the formula we have: An = 3(n – 1) + 1