Transcription of Math 55: Discrete Mathematics
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Math 55: Discrete Mathematics
Math 55: Discrete MathematicsUC Berkeley, Fall 2011 Homework # 5, due Wednesday, February (n)be the statement that13+ 23+ +n3= (n(n+ 1)/2)2forthe positive ) What is the statementP(1)?b) Show thatP(1)is ) What is the induction hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive ) Explain why these steps show that this formula is true for allpositive )P(1) is the statement 13= ((1(1 + 1)/2) ) This is true because both sides of the equation evaluate to ) The induction hypothesis is the statementP(k) for some positiveintegerk, that is, the statement 13+ 23+ +k3= (k(k+ 1)/2) ) Assuming thatP(k) holds, we need to show thatP(k+ 1) holds,that is, we need to derive the equation 13+23+ +k3+(k+1)3=((k+ 1)(k+ 2)/2)2from the equation in (c).
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 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)? b) Show that P(1) is true. c) What is the induction hypothesis? d) What do you need to prove in the inductive step?
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