Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA I Lesson 26: Recursive Challenge—The Double and Add-5 Game The “double and add 5” game is loosely related to the Collatz conjecture—an unsolved conjecture in mathematics named after Lothar Collatz, who first proposed the problem in 1937. The conjecture includes a recurrence relation, “triple and add 1,” as part of the problem statement. A worthwhile activity for you and your class is to read about the conjecture online. Classwork Example 1 Fill in the “doubling and adding 5” below: Number Double and add 5 1 1∙2+5= 7 7 _______________ ______ _______________ ______ _______________ ______ _______________ Exercise 1 Complete the tables below for the given starting number. Number 2 _______________ ______ _______________ 3 © 2013 Common Core, Inc. Some rights reserved. commoncore.org _______________ ______ Number Lesson 26: Date: Double and add 5 Double and add 5 _______________ ______ _______________ ______ _______________ Recursive Challenge Problem—The Double and Add 5 Game 2/18/16 S.143 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 143 Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA I Exercise 2 Given a starting number, double it and add 5 to get the result of round 1. Double the result of Round 1 and add 5, and so on. The goal of the game is to find the smallest starting whole number that produces a result of 100 or greater in 3 rounds or less. Exercise 3 Using a generic initial value, 𝑎0 , and the recurrence relation, 𝑎𝑖+1 = 2𝑎𝑖 + 5, for 𝑖 ≥ 0, find a formula for 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 in terms of 𝑎0 . Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Recursive Challenge Problem—The Double and Add 5 Game 2/18/16 S.144 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 144 Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA I Problem Set 1. Write down the first 5 terms of the recursive sequences defined by the initial values and recurrence relations below: a. 𝑎0 = 0 and 𝑎𝑖+1 = 𝑎𝑖 + 1, for 𝑖 ≥ 0, b. 𝑎1 = 1 and 𝑎𝑖+1 = 𝑎𝑖 + 1, for 𝑖 ≥ 1, c. 𝑎1 = 2 and 𝑎𝑖+1 = 𝑎𝑖 + 2, for 𝑖 ≥ 1, d. 𝑎1 = 3 and 𝑎𝑖+1 = 𝑎𝑖 + 3, for 𝑖 ≥ 1, e. 𝑎1 = 2 and 𝑎𝑖+1 = 2𝑎𝑖 , for 𝑖 ≥ 1, f. 𝑎1 = 3 and 𝑎𝑖+1 = 3𝑎𝑖 , for 𝑖 ≥ 1, g. 𝑎1 = 4 and 𝑎𝑖+1 = 4𝑎𝑖 , for 𝑖 ≥ 1, h. 𝑎1 = 1 and 𝑎𝑖+1 = (−1)𝑎𝑖 , for 𝑖 ≥ 1, i. 𝑎1 = 64 and 𝑎𝑖+1 = (− )𝑎𝑖 , for 𝑖 ≥ 1, 1 2 2. Look at the sequences you created in problems 1b through 1d. How would you define a recursive sequence that generates multiples of 31? 3. Look at the sequences you created in problems 1e through 1g. How would you define a recursive sequence that generates powers of 15? 4. The following recursive sequence was generated starting with an initial value of 𝑎0 , and the recurrence relation 𝑎𝑖+1 = 3𝑎𝑖 + 1, for 𝑖 ≥ 0. Fill in the blanks of the sequence: (_______, _______, 94, _______, 850, ______). 5. For the recursive sequence generated by initial value, 𝑎0 , and recurrence relation, 𝑎𝑖+1 = 𝑎𝑖 + 2, for 𝑖 ≥ 0, find a formula for 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 in terms of 𝑎0 . Describe in words what this sequence is generating. 6. For the recursive sequence generated by initial value, 𝑎0 , and recurrence relation, 𝑎𝑖+1 = 3𝑎𝑖 + 1, for 𝑖 ≥ 0, find a formula for 𝑎1 , 𝑎2 , 𝑎3 in terms of 𝑎0 . Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Recursive Challenge Problem—The Double and Add 5 Game 2/18/16 S.145 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 145