College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 9 - Section 9.4 - Mathematical Induction - 9.4 Assess Your Understanding - Page 670: 20


See below.

Work Step by Step

Proofs using mathematical induction consists of two steps: 1) The base case: here we prove that the statement holds for the first natural number. 2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number, then we prove that then the statement also holds for $n + 1$. Hence here: 1) For $n=1: 1^3+2(1)=3$ and this is divisible by $3$. 2) Assume for $n=k: k^3+2k$ is divisible by $3$. Then for $n=k+1:(k+1)^3+2(k+1)=k^3+3k^2+3k+1+2k+2=k^3+3k^2+5k+3$ and we know that $k^3+2k$ is divisible by $3$ by the inductive hypothesis, $3k^2+3=3(k^2+1)$ is also divisible by $3$, thus their sum is also divisible by $3$. Thus we proved what we wanted to.
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