Let $u$ be the sequence defined by $u _ { 0 } = 2$ and, for every natural integer $n$, by $$u _ { n + 1 } = 2 u _ { n } + 2 n ^ { 2 } - n .$$ We also consider the sequence $v$ defined, for every natural integer $n$, by $$v _ { n } = u _ { n } + 2 n ^ { 2 } + 3 n + 5$$
Here is an extract from a spreadsheet:
A
B
C
1
$n$
$u$
$v$
2
0
2
7
3
1
4
14
4
2
9
28
5
3
24
56
6
4
63
7
What formulas were written in cells C2 and B3 and copied downward to display the terms of the sequences $u$ and $v$?
Determine, by justifying, an expression of $v _ { n }$ and of $u _ { n }$ as a function of $n$ only.
\section*{Exercise 2}
Let $u$ be the sequence defined by $u _ { 0 } = 2$ and, for every natural integer $n$, by
$$u _ { n + 1 } = 2 u _ { n } + 2 n ^ { 2 } - n .$$
We also consider the sequence $v$ defined, for every natural integer $n$, by
$$v _ { n } = u _ { n } + 2 n ^ { 2 } + 3 n + 5$$
\begin{enumerate}
\item Here is an extract from a spreadsheet:
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& A & B & C \\
\hline
1 & $n$ & $u$ & $v$ \\
\hline
2 & 0 & 2 & 7 \\
\hline
3 & 1 & 4 & 14 \\
\hline
4 & 2 & 9 & 28 \\
\hline
5 & 3 & 24 & 56 \\
\hline
6 & 4 & 63 & \\
\hline
7 & & & \\
\hline
\end{tabular}
\end{center}
What formulas were written in cells C2 and B3 and copied downward to display the terms of the sequences $u$ and $v$?
\item Determine, by justifying, an expression of $v _ { n }$ and of $u _ { n }$ as a function of $n$ only.
\end{enumerate}