Answer
See explanations.
Work Step by Step
Step 1. Prove the formula is true when $n=1$: $LHS=2^3=8$, $RHS=2\times1^2(1+1)^2=8$, thus it is true for $n=1$.
Step 2. Assume the formula is true when $n=k$: we have $2^3+4^3+6^3+...+(2k)^3=2k^2(k+1)^2$
Step 3. Prove it is true for $n=k+1$: $LHS=2^3+4^3+6^3+...+(2k)^3+(2k+2)^3=2k^2(k+1)^2+(2k+2)^3
=2(k+1)^2(k^2+4(k+1))=2(k+1)^2(k+2)^2$
Use $n=k+1$ on the right side of the formula, we have $RHS=2(k+1)^2(k+2)^2$
Thus, we have $LHS=RHS$ which proves the formula is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the formula is true for all natural numbers n.
