Answer
$100n\leq n^3$ when $n\geq10$, see explanations.
Work Step by Step
Step 1. Find the relation between $100n$ and $n^3$: Let $100n=n^3$, we can get $n=10$, and we can see that $100n\leq n^3$ when $n\geq10$
Step 2. Prove the statement is true when $n=10$: $100\times10=10^3$, thus it is true for $n=10$.
Step 3. Assume the statement is true when $n=k$ ($k\geq10$): we have $100k\leq k^3$.
Step 4. Prove it is true for $n=k+1$: $RHS-LHS=(k+1)^3-100(k+1)=k^3+3k^2+3k+1-100k-100\geq 3k^2+3k-99$. As $k\geq10$, we have $k^2\geq100$, thus $RHS-LHS\geq 300+30-99\geq 0$, or $LHS\leq RHS$
Thus, the statement is also true for $n=k+1$
Step 5. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n\geq 10$.
