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Answer Key Name: ____________________________________ Date: __________________ ARITHMETIC SEQUENCES COMMON CORE ALGEBRA I There are many types of sequences, but there is one that is related to linear functions and in fact is a type of discrete linear function. These are known as arithmetic sequences. Let’s illustrate one first. Exercise #1: Evin is saving money for to buy a new toy. She already has $12 in her account. She gets an allowance of $4 per week and plans to save $3 in her account. (a) Fill out the table below for the amount of money Evin has after n weeks of saving. n a n 1 $15 2 $18 3 $21 4 $24 5 $27 6 $30 7 $33 (c) What’s wrong with the graph of the sequence shown below? a n 34 32 30 28 26 24 22 20 18 16 14 (b) Write a recursive definition for this sequence. n a 1 15 or a1 15 1 2 3 4 5 6 7 a n a n 1 3 or an an 1 3 (d) Evin proposes the following explicit formula for the amount of savings, a, as a function of the number of weeks saved n. Is this formula correct? Test it! Test the formula for n 1 : a n 3n 15 a 1 3 1 15 3 15 18 We are only modeling the amount of money that Evin has in terms of whole weeks. To connect the points means that she will be putting money in the account continuously, where she is only putting it in once a week. This graph would also imply that we could have very strange amounts saves, such as $19.5873295. But a 1 15! This formula is incorrect. Arithmetic sequences are ones where the terms in the list increase or decrease by the same amount given a unit increase in the index (where the number is in line). Exercise #2: An arithmetic sequence is given using the recursive definition: b1 3 and bi bi 1 6 . Which of the following is the value of b4 ? Show the work that leads to your answer. (1) 24 (3) 21 (2) 12 (4) 15 b2 b1 6 3 6 3 b3 b2 6 3 6 9 b4 b3 6 9 6 15 COMMON CORE ALGEBRA I, UNIT #4 – LINEAR FUNCTIONS AND ARITHMETIC SEQUENCES – LESSON #13 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013 (4) Arithmetic sequences are relatively easy to spot and are easy to fill in, so to speak. Exercise #3: For each of the following sequences, determine if it is arithmetic based on the information given. If it is arithmetic, fill in the missing blank. If it is not, show why. 17 (a) 5, 9, 13, _________, 21, 25 4 4 4 4 (b) 5, 10, 20, 40, __________, 160 This is not an arithmetic sequence because the first 5 10 two differences are not equal. As soon as we add a different amount, we know it is not arithmetic. 4 2 5 , 8 (c) 7, 4, 1, __________, 3 3 3 3 (d) 64, 16, 4, ___________, 48 12 3 1 1 , 4 16 This is not an arithmetic sequence because the first two differences are not equal. As soon as we add a different amount, we know it is not arithmetic. Finding a specific term in an arithmetic sequence without listing them sometimes can be a challenge, but not if you take your time and really think about it. Exercise #4: Consider an arithmetic sequence whose first three terms are given by: 4, 14, 24 (a) What is the 4th term? How many times was 10 added to 4 to get to the 4th term? Show a diagram to illustrate this. (b) Use what you learned in part (a) to find the value of a10 , the 10 th term. a 4 24 10 34 a10 will have 10 added on 9 times: a 4 4 10 10 10 a10 4 9 10 4 90 10 was added on three times! a10 94 (c) Write a recursive formula for the an based on the term number n. (d) Write an explicit formula for an . an 4 n 1 10 a1 4 or an an 1 10 an 4 10 n 1 Exercise #5: Seats in a small amphitheater follow a pattern where each row has a set number of seats more than the last row. If the first row has 6 seats and the fourth row has 18, how many seats does the last row, which is the 20th, have in it? Show your work to justify your response. 6 6 x 1st Row 2nd Row x 6 2x 3rd Row x 6 3 x 18 4th Row x 6 3 x 18 6 3 x 6 18 6 Now for the 20th row: 3 x 12 a 20 6 19 4 3 x 12 3 3 a 20 6 76 x4 We add 4 to get to each new row. a 20 82 82 seats in the 20th row. COMMON CORE ALGEBRA I, UNIT #4 – LINEAR FUNCTIONS AND ARITHMETIC SEQUENCES – LESSON #13 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013 Answer Key Name: ____________________________________ Date: __________________ ARITHMETIC SEQUENCES COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. An arithmetic sequence is given using the recursive definition: b1 8 and bi bi1 2 . Which of the following is the value of b4 ? Show the work that leads to your answer. (1) 14 (3) 6 (2) 2 (4) 4 b2 b1 2 8 2 6 b3 b2 2 6 2 4 b4 b3 2 4 2 2 (2) 2. For each of the following sequences, determine if it is arithmetic based on the information given. If it is arithmetic, fill in the missing blank. If it is not, show why. 48 (a) 12, 24, 36, _________, 60, 72 12 12 12 12 12 28 (c) __________, 24, 20, 16, 12, 8 4 4 4 4 (b) 10000, 1000, __________, 10, 1 9 9000 This is not an arithmetic sequence because these two differences are unequal. (d) 3 1 1 5 4 , , ________, 1, 4 2 4 4 1 4 1 4 1 4 1 4 3. Given a sequence defined by the explicit formula g n 15n 35, write out the first four terms. Then, create a recursive definition and graph the sequence on the interval 1 n 7 . a n g 1 15 1 35 15 35 50 g 2 15 2 35 20 35 65 g 3 15 3 35 45 35 80 g 4 15 4 35 60 35 95 Recursive Definition: g 1 50 g n g n 1 15 150 140 130 120 110 100 90 80 70 60 50 n 1 2 3 4 5 6 7 COMMON CORE ALGEBRA I, UNIT #4 – LINEAR FUNCTIONS AND ARITHMETIC SEQUENCES – LESSON #13 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013 4. Which of the following is an arithmetic sequence? (1) 2, 4, 8, 16, 32, 64 (3) 1, 1, 2, 3, 5, 8, 13 (2) 50, 45, 40, 35, 30 (4) 1, 1 1 1 1 , , , 2 4 8 16 All terms in (2) are 5 less than the previous term. In all other sequences, this difference is not constant. (2) APPLICATIONS 5. Evin is building a tower out of paper cups. In each row (counting from the floor up), there are two less cups than the row below it. The first row has 26 cups in it. (b) Give a recursive definition for this arithmetic (a) State the number of cups in the second, third, sequence. Recursive Definition: and fourth rows. 2nd Row: 24 cups 3rd Row: 22 cups 4th Row: 20 cups r 1 26 r n r n 1 2 (c) How many cups will be in the 11th row? Show the calculation that leads to your answer. For the 11th row we will subtract 2 only 10-times: r 11 26 10 2 26 20 6 REASONING 6. Eric considers a sequence of numbers given by the following definition b1 7 and bi bi 1 4 and decides the first 4 numbers are: 4, 11, 18, 25 (a) Interpret in your own words, what the sequence is saying and what he actually did. The sequence definition is saying that the first term is 7 and each terms is 4 more than the last term. Eric, however, interpreted the recursive rule as the first term being 4 and each term being 7 more than the last term. (b) What should the first four numbers be? b1 7 b2 b1 4 7 4 11 b3 b2 4 11 4 15 7, 11, 15, 19 b4 b3 4 15 4 19 COMMON CORE ALGEBRA I, UNIT #4 – LINEAR FUNCTIONS AND ARITHMETIC SEQUENCES – LESSON #13 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013