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).
5.3.6 Determine whether each of these proposed de nitions is a valid recur- sive de nition of a function f from the set of all nonnegative integers to the set of integers.
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