(For candidates who have NOT followed the specialization course) We consider the sequence $(u_n)$ defined by $$u_0 = 0 \quad \text{and, for every natural integer } n, u_{n+1} = u_n + 2n + 2.$$
Calculate $u_1$ and $u_2$.
We consider the following two algorithms:
\multicolumn{2}{|l|}{Algorithm 1}
\multicolumn{2}{|l|}{Algorithm 2}
Variables :
$n$ is a natural integer $u$ is a real number
Variables :
$n$ is a natural integer $u$ is a real number
Input : Processing:
Input : Processing:
\begin{tabular}{l} Enter the value of $n$ $u$ takes the value 0
For $i$ ranging from 1 to $n$ : $u$ takes the value $u + 2i + 2$
& &
Enter the value of $n$ $u$ takes the value 0
For $i$ ranging from 0 to $n - 1$ : $u$ takes the value $u + 2i + 2$
\hline & End For & & End For \hline Output : & Display $u$ & Output : & Display $u$ \hline \end{tabular} Of these two algorithms, which one allows the output to display the value of $u_n$, with the value of the natural integer $n$ being entered by the user?
Using the algorithm, we obtained the table and the scatter plot below where $n$ is on the horizontal axis and $u_n$ is on the vertical axis.
$n$
$u_n$
0
0
1
2
2
6
3
12
4
20
5
30
6
42
7
56
8
72
9
90
10
110
11
132
12
156
a. What conjecture can be made about the direction of variation of the sequence $\left( u_n \right)$? Prove this conjecture. b. The parabolic shape of the scatter plot leads to conjecturing the existence of three real numbers $a, b$ and $c$ such that, for every natural integer $n, u_n = an^2 + bn + c$. Within the framework of this conjecture, find the values of $a, b$ and $c$ using the information provided.
We define, for every natural integer $n$, the sequence $\left( v_n \right)$ by: $v_n = u_{n+1} - u_n$. a. Express $v_n$ as a function of the natural integer $n$. What is the nature of the sequence $\left( v_n \right)$? b. We define, for every natural integer $n, S_n = \sum_{k=0}^{n} v_k = v_0 + v_1 + \cdots + v_n$. Prove that, for every natural integer $n, S_n = (n+1)(n+2)$. c. Prove that, for every natural integer $n, S_n = u_{n+1} - u_0$, then express $u_n$ as a function of $n$.
(For candidates who have NOT followed the specialization course)
We consider the sequence $(u_n)$ defined by
$$u_0 = 0 \quad \text{and, for every natural integer } n, u_{n+1} = u_n + 2n + 2.$$
\begin{enumerate}
\item Calculate $u_1$ and $u_2$.
\item We consider the following two algorithms:
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{Algorithm 1} & \multicolumn{2}{|l|}{Algorithm 2} \\
\hline
Variables : & $n$ is a natural integer $u$ is a real number & Variables : & $n$ is a natural integer $u$ is a real number \\
\hline
Input : Processing: & & Input : Processing: & \\
\hline
& \begin{tabular}{l}
Enter the value of $n$ $u$ takes the value 0 \\
For $i$ ranging from 1 to $n$ : $u$ takes the value $u + 2i + 2$ \\
\end{tabular} & & \begin{tabular}{l}
Enter the value of $n$ $u$ takes the value 0 \\
For $i$ ranging from 0 to $n - 1$ : $u$ takes the value $u + 2i + 2$ \\
\end{tabular} \\
\hline
& End For & & End For \\
\hline
Output : & Display $u$ & Output : & Display $u$ \\
\hline
\end{tabular}
\end{center}
Of these two algorithms, which one allows the output to display the value of $u_n$, with the value of the natural integer $n$ being entered by the user?\\
\item Using the algorithm, we obtained the table and the scatter plot below where $n$ is on the horizontal axis and $u_n$ is on the vertical axis.
\begin{center}
\begin{tabular}{ | c | c | }
\hline
$n$ & $u_n$ \\
\hline
0 & 0 \\
\hline
1 & 2 \\
\hline
2 & 6 \\
\hline
3 & 12 \\
\hline
4 & 20 \\
\hline
5 & 30 \\
\hline
6 & 42 \\
\hline
7 & 56 \\
\hline
8 & 72 \\
\hline
9 & 90 \\
\hline
10 & 110 \\
\hline
11 & 132 \\
\hline
12 & 156 \\
\hline
\end{tabular}
\end{center}
a. What conjecture can be made about the direction of variation of the sequence $\left( u_n \right)$? Prove this conjecture.\\
b. The parabolic shape of the scatter plot leads to conjecturing the existence of three real numbers $a, b$ and $c$ such that, for every natural integer $n, u_n = an^2 + bn + c$.\\
Within the framework of this conjecture, find the values of $a, b$ and $c$ using the information provided.\\
\item We define, for every natural integer $n$, the sequence $\left( v_n \right)$ by: $v_n = u_{n+1} - u_n$.\\
a. Express $v_n$ as a function of the natural integer $n$. What is the nature of the sequence $\left( v_n \right)$?\\
b. We define, for every natural integer $n, S_n = \sum_{k=0}^{n} v_k = v_0 + v_1 + \cdots + v_n$.\\
Prove that, for every natural integer $n, S_n = (n+1)(n+2)$.\\
c. Prove that, for every natural integer $n, S_n = u_{n+1} - u_0$, then express $u_n$ as a function of $n$.
\end{enumerate}