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Name: __________________________________ Date: ______________________ Lesson 1 – Intro to Module 3 Think about it--What is the next number in the sequence 2, 4, 6, 8, …? Other ideas: Arithmetic Sequences A sequence is an ordered list of numbers. Each number of a sequence is called a term (or element) of the sequence. An arithmetic sequence follows a pattern of adding a fixed amount from one term to the next. An explicit formula describes the nth term of a sequence using the number n. For example, if you wanted to know the 88th term, just substitute 88 for π and evaluate. Example: 1. A sequence has an explicit formula ππ = 3π − 2. What are the first 5 terms of the sequence? What is the common difference between terms? A recursive formula states the first term of the sequence and relates subsequent terms to the one before. For example, if you wanted to know the 88th term, you must know the 87th term to find it. Example: 2. State the first 4 terms of the sequence: π1 = 12 ππ = ππ−1 + 5 What is the common difference? 3. Is the sequence 1, 4, 9, 16, 25 and arithmetic sequence? Justify your reasoning. 4. What is the 100th term of the arithmetic sequence 6, 11, 16, … ? 5. What is the explicit formula for the nth term of the sequence 1, 3, 5, 7, … ? 6. Write a recursive formula for the sequence 1, 3, 5, 7, … Check: For a and b: Which type of formula is this? How do you know? a.) π΄(π+1)=π΄(π)+15 7. What is the missing term of the arithmetic sequence …-7 , 15, ___ , 59, … ? b.) π΄(π) = 15 – 3(π−1) Name _______________________________ Sequences Arithmetic Show all your work 1. What is a formula for the nth term of sequence B shown below? 1) 2) 3) 4) 2. What is the common difference of the arithmetic sequence 5, 8, 11, 14? 3. Which arithmetic sequence has a common difference of 4? 1) 4, 16, 64, 256 2) 8, 4, 0, –4 3) 6, 10, 14, 18 4) 4, 8, 14, 18 4. Find the first four terms of the recursive sequence defined below. 5. Find the third term in the recursive sequence ππ = ππ−1 − 1, where π1 = 3. 6. What are the first five terms of the sequence ππ = 4π − 7. What is the common difference? 7. State the first four terms of the sequence: π1 = 23 ππ = ππ−1 − 9 What is the common difference? 8. What is the 50th term of the arithmetic sequence 8, 14, 20, … ? 9. What is the explicit formula for the nth term of the sequence 10, 12.5, 15, 17.5 … ? 10. Write a recursive formula for the sequence 10, 8, 6, 4, … 11. What is the missing term of the arithmetic sequence …-19, 18, _____ , 92, … ? Name: ___________________________________ Date: _________________ Ticket out the door Lesson 1 Consider the sequence given by a “plus 8” pattern: 2,10,18,26, …. Shae says that the formula for the sequence is an=8π+2. Marcus tells Shae that she is wrong because the formula for the sequence is an =8π−6. a. Which formula generates the sequence by starting at π=1? At π=0? b. Find the 100th term in the sequence. Name: ___________________________________ Date: _________________ Ticket out the door Lesson 1 Consider the sequence given by a “plus 8” pattern: 2,10,18,26, …. Shae says that the formula for the sequence is an=8π+2. Marcus tells Shae that she is wrong because the formula for the sequence is an =8π−6. a. Which formula generates the sequence by starting at π=1? At π=0? b. Find the 100th term in the sequence. ****Teacher’s notes – optional problems Problem set: 1. Consider a sequence generated by the formula f(π)= 6π−4 starting with π=1. Generate the terms f(1),f(2),π(3),π(4), and π(5). In problems 2-5, for each of the following sequences: a. Write a formula for the πth term of the sequence. Be sure to specify what value of π your formula starts with. b. Using the formula, find the 15th term of the sequence. 2. The sequence follows a “plus 2” pattern: 3,5,7,9, …. a. Write a formula b. Using the formula, find the 15th term 3. The sequence follows a “times 4” pattern: 1,4,16,64, …. a. Write a formula b. Using the formula, find the 15th term 4. The sequence follows a “times -1” pattern: 6,–6,6,–6, …. a. Write a formula b. Using the formula, find the 15th term 5. The sequence follows a “minus 3” pattern: 12,9,6,3, …. a. Write a formula b. Using the formula, find the 15th term