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\begin{document}
\module{Arithmetic sequences}{02}{08}{4 periods}
\en
\nocouleur
\prereq{}
\object{\begin{itemize}
\item Discover the concept of arithmetic sequence.
\item Find out the main formulae about arithmetic sequences.
\end{itemize}}
\mater{\begin{itemize}
\item Answer sheet for the team work.
\item Lesson about arithmetic sequences.
\item Exercises about arithmetic sequences.
\item Terms from seven different sequences.
\item Beamer
\end{itemize}}
\modpart{Matching game}{10 mins}
Papers with numbers are handed out to the class. Students mingle to find the other numbers that could be part of the same sequence. First terms are specified with a star on the paper.
\modpart{Team work}{45 mins}
Working in the teams from the previous part, students have to fill an answer sheet about their sequence.
\modpart{Lesson}{30 mins}
The main results about arithmetic sequences are shown with a beamer.
\modpart{Exercises}{Remaining time}
Exercises about arithmetic sequences have to be done in groups of 3 or 4 students.
\pagebreak
\moddocdis{Arithmetic sequences}{02}{08}{Answer sheet}
\begin{enumerate}
\item Write in the cells below the first five terms of your sequence, in the correct order.
\begin{center}
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|}
\hline
& & & & \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Write in the cells below the next five terms of your sequence.
\begin{center}
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|}
\hline
& & & & \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Write in the cells below the terms of your sequence whose indices are given. For example, in the first cell you have to write the $20$th term in your sequence.
\begin{center}
\begin{tabular}{|c|p{2cm}|c|p{2cm}|c|p{2cm}|c|p{2cm}|}
\hline
$20$ & & $25$ & & $50$ & & $100$& \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item What is the common difference $d$ between two consecutive terms in your sequence ?
\begin{center}
\begin{tabular}{|p{2cm}|}
\hline
$d=$ \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Let's note $a_1$ the first term in your sequence, $a_2$ the second term and so on. Give the \emph{notation} -- not the value -- of the term next to each of the terms below.
\begin{center}
\begin{tabular}{|c|p{2cm}|c|p{2cm}|c|p{2cm}|c|p{2cm}|}
\hline
$a_6$ & & $a_{12}$ & & $a_{153}$ & & $a_{n}$ & \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between any term $a_n$ and the next term.
\begin{center}
\begin{tabular}{|p{6cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between a term $a_n$, the common difference $d$ and the first term $a_1$.
\begin{center}
\begin{tabular}{|p{6cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Use the formula you found in the previous question to compute directly these terms.
\begin{center}
\begin{tabular}{|c|p{2cm}|c|p{2cm}|c|p{2cm}|c|p{2cm}|}
\hline
$a_{200}$ & & $a_{250}$ & & $a_{500}$ & & $a_{1000}$ & \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between any term $a_n$ and the term $a_2$.
\begin{center}
\begin{tabular}{|p{6cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between any term $a_n$ and the term $a_5$.
\begin{center}
\begin{tabular}{|p{6cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between any two terms $a_n$ and $a_m$.
\begin{center}
\begin{tabular}{|p{6cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\pagebreak
\item Place the first ten terms of your sequence on the graph below. To do so, choose a convenient scale on the $y$-axis.
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(0,-1)(9,9)
\psgrid[gridlabels=0](0,0)(9,9)
\newcounter{num}
\newcommand{\num}{\stepcounter{num}\arabic{num}}
\multido{\n=0+1}{10}{
\uput[d](\n,0){\num}
}
\end{pspicture}
\end{center}
\item What do you notice about the graph of this sequence ?
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Compute the sum of the first five terms of your sequence.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between the sum of the first five terms, the number of terms, the first term and the fifth term.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\rule{0pt}{15pt}\\[25pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between the sum of the first ten terms, the number of terms, the first term and the tenth term.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\rule{0pt}{15pt}\\[25pt]
\hline
\end{tabular}
\end{center}
\item Find a relation between the sum of the first $n$ terms, the number of terms, the first term and the $n$-th term.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\rule{0pt}{15pt}\\[25pt]
\hline
\end{tabular}
\end{center}
\end{enumerate}
\pagebreak
\moddocdis{Arithmetic sequences}{02}{08}{Lesson}
\section{Definition and criterion}
\txt{A sequence of numbers $(a_n)$ is arithmetic if the difference
between two consecutive terms is a constant number. Intuitively, to
go from one term to the next one, we always add the same number.}
\defi{Arithmetic sequence}{A sequence of numbers $(a_n)$ is arithmetic if, for any
positive integer $n$, $a_{n+1}-a_n=d$ where $d$ is a fixed real
number, called the \emph{common difference} of the sequence. We can
also write that $$a_{n+1}=a_n+d.$$ This equality is called the
\emph{recurrence relation} of the sequence.}
\prop{Graph of an arithmetic sequence}{The graph of an arithmetic sequence is a straight line.}
\section{Relations between terms}
\prop{Explicit definition}{For any positive integer $n$, $$a_n=a_1+(n-1)\times d.$$ This
equality is called the \emph{explicit definition} of the sequence.}
\dem{First, this equality is true when $n=1$, as $a_1=a_1+0\times
d=a_1+(1-1)\times d$. Then, suppose that it is true for a value
$n=k$, meaning that $a_k=a_1+(k-1)\times d$. Then, from the
definition of the sequence, $a_{k+1}=a_k+d=a_1+(k-1)\times
d+d=a_1+k\times d$. So the formula is true for $n=k+1$ too. So it's
true for $n=0$, $n=1$, $n=2$, $n=3$, etc, for all values of $n$.}
\prop{Relation between two terms}{For any two positive integers $n$ and $m$,
$$a_n=a_m+(n-m)\times d.$$}
\dem{From the explicit definition of the sequence $(a_n)$,
$a_n=a_1+(n-1)\times d$ and $a_m=a_1+(m-1)\times d$, so
$a_n-a_m=(a_1+(n-1)\times d)-(a_1+(m-1)\times d)=n\times d-m\times d
=(n-m)d$. Therefore, $a_n=a_m+(n-m)\times d$.}
\pagebreak
\thispagestyle{empty}
\section{Limit when $n$ approaches $+\infty$}
\theo{Limit of an arithmetic sequence}{The limit of an arithmetic sequence $(a_n)$ of common
difference $d$
\begin{itemize}
\itemb is equal to $+\infty$ when $d>0$ ;
\itemb is equal to $-\infty$ when $d<0$ ;
\itemb is equal to $a_1$, trivially, if $d=0$.
\end{itemize}}
\dem{
\begin{itemize}
\itemb Suppose that $a_1>0$ and $d>0$, and consider any real number $K$.
Then, the inequation $a_n>K$, or $a_1+(n-1)d>K$, is solved by any
positive integer $n$ such that $n>\frac{K-a_1}{d}+1$. This means
that for any real number $K$, there exist some integer $N$ such that
for any $n\geq N$, $a_N>K$. This is exactly the definition of the
fact that $\lim a_n=+\infty$.
\itemb Suppose that $a_1>0$ and $d<0$, and consider any real number $K$.
Then, the inequation $a_n\frac{K-a_1}{d}+1$. This means
that for any real number $K$, there exist some integer $N$ such that
for any $n\geq N$, $a_N