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

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.

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

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)

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

Continued….. Input (term) Output (value of the term) 1 2 4 3 7 10 5 13 6 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

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?

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.

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