EECS 1019 Study Guide - Midterm Guide: Rational Number, Natural Number, Pigeonhole Principle

• School: York University
• Department: Electrical Engineering and Computer Science
• Course: EECS 1019
• Professor: Suprakash Datta

Document Summary

[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.