Prove by induction that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$.

Question

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

Answer 1

Hints:

$$1^3=\frac{1^2\cdot2^2}4\;\;\color{green}\checkmark$$

$$1^3+2^3+\ldots+n^3+(n+1)^3\stackrel{\text{Ind. Hyp.}}=\frac{n^2(n+1)^2}4+(n+1)^3=$$

$$=\frac{(n+1)^2}4\left(n^2+4(n+1)\right)=\ldots\ldots\;\;\;\color{green}\checkmark$$

Answer 2

Here's the general idea behind the Principle of Mathematical Induction, a la a staircase metaphor:

(1) Show that the bottom step of the staircase is painted.

(2) Show that, if a given step of the staircase is painted, then the next step up is painted, too.

PMI lets us conclude that all of the steps are painted. Is this a visually convincing metaphor?

In light of your example, it seems you've completed step 2. All that's left is to complete step 1--that is, show that $$1^3=\frac{1^2\cdot 2^2}4.$$