<s>[INST] Prove $n^3+(n+1)^3>(n+2)^3$ Here is the question: for which natural numbers do the following inequalities hold true? State a claim and prove
it:
$n^3+(n+1)^3>(n+2)^3$.
I think this statement holds when $n\ge6$, where $n^3+(n+1)^3=559$, and $(n+2)^3=512$.
I thought about using induction, but I couldn't express $(n+1)^3+(n+2)^3$ in terms of the inductive hypothesis-  $n^3+(n+1)^3$.
So I'm wondering if there's a better way than induction to prove it or how to use the inductive hypothesis in a useful way given that induction is a good way to prove this.
 [/INST] Need to show:
$n^3\gt \dfrac{(n+2)^3-(n+1)3}{1}=$
$=3(t^2)$, $t \in (n+1,n+2)$
(MVT)
Hence:
$3(n+2)^2 \gt (n+2)^3-(n+1)^3$.
For $n \ge 6:$
$n^3 > 3(n+2)^2$.
Since:
The inequality is true for $n=6$.
$f(x):= x^3 -3(x+2)^2$.
$f'(x) = 3x^2 -6(x+2) =$
$3(x^2-2x -4) = $
$3[( x-1)^2 -5] >0$, for $x \ge 6$,
hence $f(x)$ is strictly increasing 
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