COMMON CORE STANDARDS for MATHEMATICS FUNCTIONS: INTERPRETING FUNCTIONS (F-IF) F-IF3. Recognize that sequences are functions, sometimes defined recursively. Whose domain is a subset of the integers. FUNCTIONS: BUILDING FUNCTIONS (F-BF) F-BF2. Write an arithmetic and geometric sequences both recursively and with explicit formula, use them to model situations and translate between the two forms. FUNCTIONS: LINEAR, QUADRATIC, AND EXPONENTIAL MODELS F-LE 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or to input-output pairs (include reading from a table)

INTRO TO SEQUENCES AND SERIES Types of a “Sequence”? Arithmetic: A sequence is aritmetic if the differences between consecutive terms are the same. So the sequence a 1, a 2, a 3, a 4,... a n,... Is arithmetic if there is a number d such that a 2 -a 1 = a 3 -a 2 = a 4 -a 3 =... = d The number d is the common difference of the arithmetic sequence

Identify which of the following are arithmetic sequences? For each arithmetic sequence, find the common difference. ARITHMETIC SEQUENCES a)-4, -1, 2, 5, 8, … b)2, 5, 10, 17, 26, … c)117, 115, 113, 111, 109, … d)5, 7, 9, 13, 17,... Explain how you can tell?

Write the first five terms of the arithmetic sequences described below ARITHMETIC SEQUENCE Make a table presenting the first five terms. Plot the points on a graph, what do you notice?

General Formula for the nth term of an Arithmetic Sequence Given the sequence 3, 5, 7, 9, … What is the common difference (we will call that d)? To get from the 1 st term to the 4 th term, how many times did you have to add d to the first term? To get from the 1 st term to the 7 th term, how many times did you have to add d to the first term? Find the next 4 terms.11, 13, 15, To get from the 1 st term to the 100 th term, how many times would you have to add d to the first term? 99

Based on this information, what is the 100 th term? 201 Write an expression that for the 100 th term (we will call it a 100 ). a 100 = (2) Let’s generalize this : a n = a 1 + (n - 1)d Let’s write a formula for this particular sequence: a n = 3 + (n - 1)2 a n = 3 + 2n - 2 a n = 2n + 1

Now let’s look at this another way: The first term was 3, the second term was 5, …, the 8 th term was 17. (1, 3), (2, 5), …, (8, 17) Let’s pair up the term number with the term itself. If we look at these as ordered pairs, what is the slope between any two of these points? The slope has the same value as what? d

Let’s write an equation of the line in slope intercept form This is the exact same equation we got using the other formula!

Use either method to write a rule for each sequence below. Then find the 25 th term in the sequence. a) 6, 14, 22, 30, 38, … b) 10, 7, 4, 1, …

Use either method to write a rule for each sequence below. Then find the 25 th term in the sequence.