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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 a1, a2, a3, a4,. . . an, . . . Is arithmetic if there is a number d such that a2-a1= a3-a2= a4-a3= . . . = d The number d is the common difference of the arithmetic sequence ARITHMETIC SEQUENCES Identify which of the following are arithmetic sequences? For each arithmetic sequence, find the common difference. a) b) c) d) -4, -1, 2, 5, 8, … 2, 5, 10, 17, 26, … 117, 115, 113, 111, 109, … 5, 7, 9, 13, 17, ... Explain how you can tell? ARITHMETIC SEQUENCE Write the first five terms of the arithmetic sequences described below an 3 2n 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, … Find the next 4 terms. 11, 13, 15, 17 What is the common difference (we will call that d)? 2 To get from the 1st term to the 4th term, how 3 many times did you have to add d to the first term? To get from the 1st term to the 7th term, how 6 many times did you have to add d to the first term? To get from the 1st term to the 100th term, 99 how many times would you have to add d to the first term? Based on this information, what is the 100th term? 201 Write an expression that for the 100th term (we will call it a100 ). a100 = 3 + 99 (2) Let’s generalize this: an = a1 + (n - 1)d Let’s write a formula for this particular sequence: an = 3 + (n - 1)2 an = 3 + 2n - 2 an = 2n + 1 Now let’s look at this another way: The first term was 3, the second term was 5, …, the 8th term was 17. Let’s pair up the term number with the term itself. (1, 3), (2, 5), …, (8, 17) If we look at these as ordered pairs, what is the slope between any two of these points? 17 3 14 2 8 1 7 The slope has the same value as what? d Let’s write an equation of the line in slope intercept form y 2 x b substituting a point in for x and y 17 2(8) b 17 16 b 1 b so y 2 x 1 or a n 2n 1 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 25th 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 25th term in the sequence. a. a1 5 d 3 b. a7 10 d 3