EECS 1019 Study Guide - Midterm Guide: Rational Number, Natural Number, Pigeonhole Principle
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EECS 1019 Full Course Notes
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[4 points] prove using mathematical induction that for all natural numbers n, Inductive step: assume the statement is true for n = k. then for n = k + 1, 1 k + 1 (cid:112)k(k + 1) + 1 (cid:112)k2) + 1 k + 1 k + 1 k + 1 k + 1 k + 1. [4 points] prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9. Solution: we prove this by using induction on n. True, since 13 + 23 + 33 = 1 + 8 + 27 = 36 is divisible by 9. Inductive step: assume the statement is true for n = k. so k3 + (k + 1)3 + (k + 2)3 = 9a for. Then for n = k + 1, (k + 1)3 + (k + 2)3 + (k + 3)3 = (k + 1)3 + (k + 2)3 + (k + 3)3 + k3 k3.