Consider the sequence $(u_n)$ defined by:

$$\left\{ \begin{aligned}
u _ { 0 } & = 1 \text{ and, for every natural number } n, \\
u _ { n + 1 } & = \left( \frac { n + 1 } { 2 n + 4 } \right) u _ { n } .
\end{aligned} \right.$$

Define the sequence $(v_n)$ by: for every natural number $n$, $v_n = (n + 1) u_n$.

\begin{enumerate}
  \item The spreadsheet below presents the values of the first terms of the sequences $(u_n)$ and $(v_n)$, rounded to the hundred-thousandth.\\
What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_n)$?

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
 & A & B & C \\
\hline
1 & $n$ & $u_n$ & $v_n$ \\
\hline
2 & 0 & 1.00000 & 1.00000 \\
\hline
3 & 1 & 0.25000 & 0.50000 \\
\hline
4 & 2 & 0.08333 & 0.25000 \\
\hline
5 & 3 & 0.03125 & 0.12500 \\
\hline
6 & 4 & 0.01250 & 0.06250 \\
\hline
7 & 5 & 0.00521 & 0.03125 \\
\hline
8 & 6 & 0.00223 & 0.01563 \\
\hline
9 & 7 & 0.00098 & 0.00781 \\
\hline
10 & 8 & 0.00043 & 0.00391 \\
\hline
11 & 9 & 0.00020 & 0.00195 \\
\hline
\end{tabular}
\end{center}

  \item a. Conjecture the expression of $v_n$ as a function of $n$.\\
b. Prove this conjecture.
  \item Determine the limit of the sequence $(u_n)$.
\end{enumerate}