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Arithmetic Sequences and Recursive Formulas Why is a sequence a function and how can you model a function for an arithmetic sequence? F.IF.2, F.IF.3, F.BF.1, F.BF.2, F.LE.2 Sequence – an ordered list of numbers or items. Each element in the sequence is called a term. Terms can be paired with a position number (n) – this creates a function where the domain is set by the n’s and the range is set by the terms. The position numbers (n) are consecutive integers that usually start with 0 or 1. Ex 1 – Given a sequence f (2, 5, 8, 11, 14…), and the domain (position) starts at 0, what is f(4)? Explain what it represents and how you determined your answer. How would your answers change if the domain started with 1? Ex 2 – Predict the next term in the sequence: 50, 42, 34, 26, 18,… Explain your reasoning. Let’s discuss an example of a sequence found in our everyday lives. What makes our example a sequence? Some sequences can be described using an algebraic model. An explicit rule allows you to find the n-th term of the function. Ex 3 – Find the first 4 terms of a sequence that follows the explicit rule f(n) = 3n + 2, assuming the domain starts with n = 1. What do you notice about the terms and how is it related to the equation? ◦ What would the 10th term be? Sometimes we can find subsequent terms in a sequence by knowing what the previous terms are - we use recursive rules to find the n-th term based on previous terms. Ex 4 – Given that f(1) = 3 and f(n) = f(n-1) + 2 for n≥2, find the first 4 terms. n f(n-1) + 2 f(n) 1 Given 3 2 3 4 Describe how you would find the 12th term? How would you find the 50th term? Is there a more effective way than the recursive model? If the difference between each term is consistent, the function is an arithmetic sequence, where d is the common difference. Ex 5 – The table shows end-of-month bank balances for an account that does not earn interest. Month n 1 2 3 4 5 Balanc e f(n) 60 80 100 120 140 a) What is the common difference? Write the recursive rule, given f(1) = 60. b) Write the explicit rule for the above arithmetic sequence. b) Making a table may help with the explicit form. n f(n) 1 60 + 20 (1-1) = 60 + 20(0) = 60 2 3 4 5 The common difference is related to something else you are familiar with – what? Ex 6 – If the only things you know about a particular arithmetic sequence is that the common difference is 4 and the 3rd term is 10, how can you find the 51st term? One way to find a rule that will work for any all arithmetic sequences, is to look at a specific sequence then make generalities. Let’s use 6, 9, 12, 15, 18, … Specific General/Algebra Common difference d= d Recursive Rule Given f(1) = 6 Given f(1) f(n) = f(n -1) + Explicit Rule f(n) = + (n-1) for n≥2 f(n) = f(n-1) + for n≥2 f(n) = + (n-1) Ex 7 – a) If f(1)= 4 and d = 10, how can you find the 7th term of the sequence? b) What information do you need to know in order to find the 11th term of an arithmetic sequence using a recursive model? c) With a partner, discuss a real life situation when finding terms of an arithmetic sequence would be useful. Ex 8 – The graph shows how the cost of a snowboarding trip depends on the number of boarders. Make a chart of the data. Axis Title Cost ($) 200 n 150 1 100 2 50 Cost ($) 0 0 5 Axis Title 3 4 f(n) a) Is the sequence arithmetic? Why or why not? b) Write the explicit rule. c) Using the graph on the last slide, write an equation of line using slope-intercept form. d) What do you notice about the answers in b and c? Explain. (use m and d in your response). With your partner, come up with the pattern for the famous Fibonacci Sequence (hint, it is NOT an arithmetic sequence, but you can still find an answer). 1, 1, 2, 3, 5, 8, 13, … Let’s develop the recursive rule together: