Use Mathematical Induction to prove equation? Use mathematical induction to prove each of the following statements.
let $$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$
Show that the function $$g(n)= \frac{n^2(n+1)^2}{4}$$ for all n in N
so the base case is just g(1) right? so the answer for the base case is 1, because 4/4 = 1
then for g(2) is it replace all of the n's with n + 1 and see if there is a concrete answer?

Show that: $$1^3 + 2^3 + 3^3 + ... + n^3= \frac{n^2(n+1)^2}{4}$$

$(1)$ First step is to evaluate the expression for $n=1$, $$1^3=\frac{1\times(1+1)^2}{4}$$
Which is $1=1$, which means we can proceed.

$(2)$ Now assume it is true for $n$, $$1^3 + 2^3 + 3^3 + ... + n^3= \frac{n^2(n+1)^2}{4}$$

$(3)$ Lastly prove for $n+1$ and you have proved it for all $n\in\mathbb N$,
$$1^3 + 2^3 + 3^3 + ... +n^3+ (n+1)^3= \frac{(n+1)^2(n+2)^2}{4}$$
After "inserting" $n+1$, replace the left side as it follows:
$$ \frac{n^2(n+1)^2}{4}+(n+1)^3=\frac{(n+1)^2(n+2)^2}{4}$$
Simply "tidy up" the left side to get the final form;
$$\frac{(n+1)^2(n+2)^2}{4}=\frac{(n+1)^2(n+2)^2}{4}$$

And that's an example of a proof with induction.
