Alg2_M7_7.10
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San Francisco State University *
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113
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Mathematics
Date
Feb 20, 2024
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Uploaded by ConstableStingray1557
Module 7, Proofs by Mathematical Inductions Assignment
You know that a mathematical proof is a sequence of statements that demonstrates the truth of an assertion. Previously you wrote proofs using a two-column format, and you have had practice writing paragraph proofs. In this module you learned about proof by mathematical induction, which has three steps. You must show all three steps to have a complete proof. •
Step 1: Show that the formula works for a certain value.
•
Step 2: Assume the formula is true for some other value, k.
•
Step 3: Prove that the formula is true for 𝑘𝑘
+ 1
.
Consider the Tower of Hanoi game described below. There are three pegs and several disks of different sizes. The disks start on the peg to the far left and are stacked according to size from largest to smallest. The object of the game is to transfer all the disks to the peg on the far right so that they are still stacked from largest to smallest. You may move one disk at a time to any of the pegs, but you may never rest a larger disk on top of a smaller disk. Use what you know to explore the concept of proof by mathematical induction by answering the questions below. 1. Renae has been playing Tower of Hanoi and has noticed that the minimum number of moves it
takes to defeat the game is related to the number of disks she must move. She has recorded her
observations below. Write an equation that describes this pattern. Show your work.
Number of Disks in the Tower Minimum Number of Moves 1 1 2 3 3 7 4 15 5 31
2. Use a proof by mathematical induction to show that your equation from question 1 applies to the minimum number of moves required to defeat the Tower of Hanoi game, based on the number of disks you must move. Think about the process of the game; and describe how your equation applies to it.
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