Mathematical induction by inequality

Question

Prove $n^3>2n-2$ for all $n∈N $

step 1: claim (1) is 1 > 0
LHS > RHS

step 2: assume claim (k) is true, that is $k^3>2k-2$

Prove claim (k + 1)
$(k+1)^3>2(k+1)-2$

LHS = $(k+1)^3$
= $k^3+3k^2+3k+1$
$>(2k-2)+3k^2+3k+1$

Then I am struck, I am not sure how to get to $2(k+1)-2$

Unless you're required to use induction, this is quite straight forward. $n \ge 2 \implies n^3> 2n > 2n-2$ . We have $n=1$ to verify separately. — Macavity, Commented Jan 30, 2018 at 16:14
Yup, I need to use induction, but I do not understand why and how it works. — skyzhuzhu, Commented Jan 30, 2018 at 16:18

Math Lover · Accepted Answer · 2018-01-30 15:35:49Z

1

Hint: $$3k^2+3k = 3k(k+1)\ge 6 >1.$$

answered Jan 30, 2018 at 15:35

Math Lover

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