# hw6 - Homework 6 Discrete Math 1 Exercise 20 b p 230 Log(2n...

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Homework 6 Discrete Math 1. Exercise 20, b) p. 230 Log(2n)/ Log(n)=log(2)= 1 2. Exercise 20, f) p. 230 3. (2n) 3 /n 3 =8 Time increases by 8 4. Exercise 20, g) p. 230 Increases by 2 n time 5. Exercise 4, p. 329 Let P(n) be the statement that 13 +23 +···+n3 = (n(n+1)/2)2 for the positive integer n. a) What is the statement P(1)? 1 3 = ((1(1 + 1)/2) 2 b) Show that P(1) is true, completing the basis step of the proof. 1³ = 1 (1(1+1)/2)² = (2/2)² = 1² = 1 Thus, P(1) is true c) What is the inductive hypothesis? P(n) for some positive integer n, that is, the statement 1 3 + 2 3 +· · ·+n 3 = (n(n+ 1)/2) 2 d) What do you need to prove in the inductive step? We assume that P(n) is true, then we need to show that P(n + 1) is also true. To do this, we need to derive the equation 1 3 +2 3 +· · ·+n 3 +(n+1) 3 =((n + 1)(n + 2)/2) 2 from the equation 1 3 + 2 3 +· · ·+n 3 = (n(n+ 1)/2) 2 e) Complete the inductive step, identifying where you use the inductive hypothesis. 1³ + ... + n³ + (n+1)³ = ((n+1)(n+1+1)/2)² = ((n+1)(n+2)/2)² 1³ + ... + n³ + (n+1)³ = (n(n+1)/2)² + (n+1)³ = n²(n+1)²/4 + (n+1)³ = (n+1)² [n²/4 + (n+1)] = (n+1)² [(n² + 4n + 4)/4] = (n+1)² (n+2)² / 4 = ((n+1)(n+2)/2)² f) Explain why these steps show that this formula is true whenever n is a positive integer. Using this proof, we have shown that if P(n) is true, then P(n+1) is also true Thus, P(n) is true for all positive integer n. 6. Exercise 32, p.330 Prove that 3 divides n 3 +2n whenever n is a positive integer.

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