Friday, February 16, 2018

proof verification - Prove or Disprove Inequality By Induction

Prove or Disprove



$\sum_{i=0}^n(2i)^3 \le (8n)^3
$




If true, prove using induction. If false, give the smallest value of n that is a counter example and the values for the left and right hand sides of the equation.



I started out with the Base Case at n = 1:



$\sum_{i=0}^1(2i)^3 = 8, 8^3 = 512
$



$8 \le 512 \therefore
$ true




Induction Hypothesis: Assume
$\sum_{i=0}^k(2i)^3 \le (8n)^3
$ is true



Induction: $\sum_{i=0}^{k+1} (2i)^3 \le (8(k+1))^3
$



$\sum_{i=0}^{k+1} (2i)^3 = \sum_{i=0}^k(2i)^3 + (2(k+1))^3
$




This is where I'm stuck in the problem right now. I'm not sure how to use the hypothesis when it's an inequality.

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