Need help on a proof

[Selena]

Hi everyone, please help me with this problem set I am doing.

(I'm not sure if the "-" after the N is supposed to be a -> typo or something)

I'm not sure what k - 1 has to do with this claim, isn't it asking for an "n" where 2^n is greater than n^3? And then put that value in a proof?

Thanks in advance.

 

[galactus]

Besides for n=1, \(\displaystyle 2^{k}>k^{3}\) when \(\displaystyle k\geq 10\)

For the induction, it has to be shown that \(\displaystyle 2^{k+1}>(k+1)^{3}\)

 

[Selena]

Besides for n=1, \(\displaystyle 2^{k}>k^{3}\) when \(\displaystyle k\geq 10\)

For the induction, it has to be shown that \(\displaystyle 2^{k+1}>(k+1)^{3}\)

Oh thanks.
Why is it k+1 in the proof and not k? Isn't k included?
And can you give an outline of what the proof should look like? Do I simply write something along the lines of "case 1, if n is in the set of natural numbers below k, then the claim fails", plug in some number for n, and then "case 2, if n is equal or greater than k, then claim stands", plug in some number for n?