\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}
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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 $a_n<K$, 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 The last situation, when $d=0$, is obvious, as all terms are equal to $a_1$.
\end{itemize}
}

\section{Sums of consecutive terms}

\theo{Sum of consecutive terms}{Let $(a_n)$ be an arithmetic sequence, The sum $S_n$ of all the terms
between $a_1$ and $a_n$, 
$S_n=a_1+a_2+\ldots+a_{n-1}+a_n$, or more precisely
$S=\sum_{i=1}^{n} a_i$, is given by the formula
$$S=n\times\frac{a_1+a_n}{2}.$$}

\dem{The trick to prove this formula is to write the sum in two
different ways, one based on the first term, the other based on the
last term. Using the formula of proposition 1.2, we have, on one
hand
\begin{eqnarray*}
S_n &=& a_1+a_2+\ldots+a_{n-1}+a_n \\
S_n &=& a_1+a_1+d+\ldots+a_1+(n-2)d+a_1+(n-1)d
\end{eqnarray*}
and on the other hand
\begin{eqnarray*}
S_n &=& a_1+a_2+\ldots+a_{n-1}+a_n \\
S_n &=& a_n-(n-1)d+a_n-(n-2)d+\ldots+a_n-d+a_n.
\end{eqnarray*}
When we add these two expression of $S_n$, all terms involving $d$ are
cancelled and we end up with as many times the sum $a_1+a_n$ are
there were of terms in $S_n$, so
\begin{eqnarray*}
2S_n &=& n(a_m+a_n)\\
S_n &=& n\times\frac{a_1+a_n}{2}.
\end{eqnarray*}
 }

\pagebreak

\moddocdis{Arithmetic sequences}{02}{08}{Exercises}

\newcounter{chap}
\setcounter{chap}{8}

\exoc{A single square is made from 4 matchsticks. Two squares in a
row needs 7 matchsticks and 3 squares in a row needs 10 matchsticks.
This process defines a sequence. Determine, for this sequence,
\begin{enumerate}
\item the first term ;
\item the common difference ;
\item the formula for the general term ;
\item how many matchsticks are in a row of 25 squares.
\end{enumerate}
\begin{center}
\begin{pspicture}(0,0)(8,2)
%\psgrid[gridcolor=gray]
\def\match{\psline(0,0)(2,0)\psellipse*(1.8,0)(0.2,0.1)}
\rput(0,0){\match} \rput{90}(2,0){\match} \rput{180}(2,2){\match}
\rput{270}(0,2){\match} \rput(2,0){\rput(0,0){\match}
\rput{90}(2,0){\match} \rput{180}(2,2){\match}}
\rput(4,0){\rput(0,0){\match} \rput{90}(2,0){\match}
\rput{180}(2,2){\match}} \rput(6,0){\rput(0,0){\match}
\rput{90}(2,0){\match} \rput{180}(2,2){\match}}
\end{pspicture}
\end{center}}



\exoc{A triangular number is a natural number such that the shape of
an equilateral triangle can be formed by that number of points. For
$n\geq 1$, let $u_n$ be the difference between the $n$-th triangular
number and the previous one.
\begin{enumerate}
\item Find the six first triangular numbers.
\item Give the 5 first terms of the sequence $(u_n)$. What kind of
sequence could it be ?
\item Write the six first triangular numbers using consecutive terms of the
sequence $(u_n)$.
\item Deduce a formula to find directly the $n$-th trianguler
number.
\item Use this formula to compute the $36$th triangular number. Is
this number famous for other reasons ?
\end{enumerate}
}

\exoc{The third term of an arithmetic sequence is -7 and the
7$^{\rm{th}}$ term is 9. Determine:
\begin{enumerate}
\item the first term $a_1$ and the common difference $d$ ;
\item the $51^{\rm{st}}$ term.
\end{enumerate}}

\exoc{In an arithmetic sequence, the first and seventh terms are
$x^2$ and $6+x-5x^2$, respectively. If the common difference is $x$,
determine the possible values of $x$.}

\exoc{The twelfth term of an arithmetic sequence is 5, and the
common difference between successive terms is 3. Determine which
term has a value of 47.}

\exoc{In a given arithmetic sequence, the first term is 2, the last
term is 29 and the sum of all the terms is 155. Find the common
difference.}

\pagebreak

\exoc{A horizontal line intersects a piece of string at four points
and divides it into five parts, as shown below.
\begin{center}
\begin{pspicture}(-1,-2)(6,2)
%\psgrid[gridcolor=gray]
\psplot[xunit=0.00556]{90}{810}{x sin}
\psline[linestyle=dashed](-1,0)(6,0)
\psdots[dotsize=5pt](1,0)(2,0)(3,0)(4,0)
\rput(0.5,1.5){\psframebox{1}} \rput(1.5,-1.5){\psframebox{2}}
\rput(2.5,1.5){\psframebox{3}} \rput(3.5,-1.5){\psframebox{4}}
\rput(4.5,1.5){\psframebox{5}}
\end{pspicture}
\end{center}
If the piece of string is intersected in this way by 19 parallel
lines, each of which intersects it at four points, find the number
of parts into which the string will be divided.}

\exoc{These two problems appear on the Rhind Papyrus. This papyrus was named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt ; it was apparently found during illegal excavations in or near the Ramesseum (the memorial temple of Pharaoh Ramesses II). The British Museum, where the papyrus is now kept, acquired it in 1864. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus.

\begin{center}
\includegraphics[scale=0.2]{images/Rhind.eps}
\end{center}


\begin{description}
\item[Problem 40.] Divide 100 hekats of barley among 5 men so that the common difference is the same and so that the sum of the two smallest is 1/7 the sum of the three largest.
\item[Problem 64.] Divide 10 hekats of barley among 10 men so that the common difference is 1/8 of a hekat of barley.
\end{description}
}



\pagebreak

\moddoc{Cards for the matching game}

\begin{center}
\includegraphics[width=15cm]{images/08-1.eps}
\end{center}

\begin{center}
\includegraphics[width=15cm]{images/08-2.eps}
\end{center}


\end{document}
