Consider a pyramid with a square base formed of identical balls stacked on top of each other:
\begin{itemize}
  \item the $1^{\text{st}}$ level, located at the highest level, is composed of 1 ball;
  \item the $2^{\mathrm{nd}}$ level, just below, is composed of 4 balls;
  \item the $3^{\mathrm{rd}}$ level has 9 balls;
  \item the $n$-th level has $n^{2}$ balls.
\end{itemize}
For any integer $n \geqslant 1$, we denote by $u_{n}$ the number of balls that make up the $n$-th level from the top of the pyramid. Thus, $u_{n} = n^{2}$.
\begin{enumerate}
  \item Calculate the total number of balls in a pyramid with 4 levels.
  \item Consider the sequence $(S_{n})$ defined for any integer $n \geqslant 1$ by
$$S_{n} = u_{1} + u_{2} + \ldots + u_{n}.$$
a. Calculate $S_{5}$ and interpret this result.\\
b. Consider the pyramid function below written incompletely in Python.\\
Copy and complete on your paper the box below so that, for any non-zero natural integer $n$, the instruction \texttt{pyramide(n)} returns the number of balls making up a pyramid with $n$ levels.
\begin{verbatim}
def pyramide(n) :
    S = 0
    for i in range(1, n+1):
        S = ...
    return...
\end{verbatim}
c. Verify that for any natural integer $n$:
$$\frac{n(n+1)(2n+1)}{6} + (n+1)^{2} = \frac{(n+1)(n+2)[2(n+1)+1]}{6}$$
d. Prove by induction that for any integer $n \geqslant 1$:
$$S_{n} = \frac{n(n+1)(2n+1)}{6}.$$
  \item A merchant wishes to arrange oranges in a pyramid with a square base. He has 200 oranges. How many oranges does he use to build the largest possible pyramid?
\end{enumerate}