<s>[INST] 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?
 [/INST] Assuming you are asking what the inductive step is, you simply need to assume that $g(n) =\frac{n^2(n+1)^2}{4}$ and then use it to evaluate g(n+1) like so:
\begin{equation}
g(n+1) = g(n) + (n+1)^3
\end{equation}
\begin{equation}
g(n+1) = \frac{n^2(n+1)^2}{4} + (n+1)^3
\end{equation}
\begin{equation}
g(n+1) = \frac{4(n+1)^3 + n^2(n+1)^2}{4}
\end{equation}
\begin{equation}
g(n+1) = \frac{(n+1)^2(4(n+1) + n^2)}{4}
\end{equation}
\begin{equation}
g(n+1) = \frac{(n+1)^2(n^2 + 4n + 4)}{4}
\end{equation}
\begin{equation}
g(n+1) = \frac{(n+1)^2(n+2)^2}{4}
\end{equation}
Q.E.D.
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