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Assignment 6 Solutions

# Assignment 6 Solutions - Assignment#6(The same questions...

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Assignment #6 (The same questions for both sessions) Due Tuesday, 3/3 Complete the following exercises from chapter 2.4: ł   2, 6 Complete the following exercises from chapter 4.1: �   4, 6, 7, 8, 10, 20, 22, 28 Chapter 2.4: Exercise 2: Answer: Chapter 2.4: Exercise 6: Answer:

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Chapter 4.1: Exercise 4: Let P(n) be the statement that 1 3 + 2 3 + 3 3 + … + n 3 = (n(n+1)/2) 2 for the positive integer n a) What is statement of P(1)? Answer: 1 3 = (1(1+1)/2) 2 b) Show that P(1) is true by completing the basis step of the proof Answer: we show P(1) by showing both sides of P(1) are equal. 1 3 = (1(1+1)/2) 2 1 = 1 c) What is inductive hypothesis? Answer: d) What do you need to prove in the inductive step? Answer: For each k 1, P(k) implies P(k+1). In other words, assuming the inductive hypothesis, we want to show e) Complete the inductive step: Answer: [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 1 2 2 3 2 2 3 2 3 3 3 3 2 2 1 2 2 1 4 2 1 4 2 1 4 4 4 1 1 4 1 1 4 1 Hypothesis Inductive by 1 2 1 1 2 1 + + = + + = + + = + + = + + + = + + + = + + + = + + + = + + + + + k k k k k k k k k k k k k k k k k k k k k k f)
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Assignment 6 Solutions - Assignment#6(The same questions...

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