Let $P(n)$ be the property: $2^n > n^3$. Let's use mathematical induction to prove that $P(n)$ is true for $n\geq10$.
Basis: $P(10): 2^{10} > 10^3 \Leftrightarrow 1024 > 1000$ which is true.
Hypothesis: $P(k): 2^k > k^3$
Inductive step: we have to prove $P(k) \Rightarrow P(k+1)$:
$2^{k+1} = 2\cdot 2^k > 2\cdot k^3 = k^3 + k^3$
Everything is clear up to here, but then it goes on like this:
$k^3 + k^3 \geq k^3 + 7\cdot k^2$
Question here: How to prove that $k^3 > 7\cdot k^2$ ? I know it's because $n\geq 10$, but how to prove it? Besides empirically - which shows it's true.
Then the proof goes on:
$k^3 + k^3 \geq k^3 + 3\cdot k^2 + 3\cdot k + 1$
Question here: I suppose it's the same "technique" like above, isn't it? If not, how to prove that:
$k^3 \geq 3\cdot k^2 + 3\cdot k + 1$ ?