<s>[INST] Prove by induction that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$. Use mathematical induction to prove that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$.
Can anyone explain? Because I have no clue where to begin. I mean, I can show that $1^3+ 2^3 +...+ (k+1)^3=\frac{(k+1)^2(k+2)^2}{4}$, but then I don't know where to go. I need further explanation to prove it.
thank you so much for help
Sincerely
 [/INST] Hints:
$$1^3=\frac{1^2\cdot2^2}4\;\;\color{green}\checkmark$$
$$1^3+2^3+\ldots+n^3+(n+1)^3\stackrel{\text{Ind. Hyp.}}=\frac{n^2(n+1)^2}4+(n+1)^3=$$
$$=\frac{(n+1)^2}4\left(n^2+4(n+1)\right)=\ldots\ldots\;\;\;\color{green}\checkmark$$
</s>