We study the evolution over time of the number of young and adult animals in a population. For any natural integer $n$, we denote by $j _ { n }$ the number of young animals after $n$ years of observation and $a _ { n }$ the number of adult animals after $n$ years of observation. At the beginning of the first year of the study, there are 200 young animals and 500 adult animals. Thus $j _ { 0 } = 200$ and $a _ { 0 } = 500$. We admit that for any natural integer $n$ we have: $$\left\{ \begin{array} { l }
j _ { n + 1 } = 0,125 j _ { n } + 0,525 a _ { n } \\
a _ { n + 1 } = 0,625 j _ { n } + 0,625 a _ { n }
\end{array} \right.$$ We introduce the following matrices: $$A = \left( \begin{array} { l l }
0,125 & 0,525 \\
0,625 & 0,625
\end{array} \right) \text { and, for any natural integer } n , U _ { n } = \binom { j _ { n } } { a _ { n } } .$$
a. Show that for any natural integer $n , U _ { n + 1 } = A \times U _ { n }$. b. Calculate the number of young animals and adult animals after one year of observation and then after two years of observation (results rounded down to the nearest unit). c. For any non-zero natural integer $n$, express $U _ { n }$ as a function of $A ^ { n }$ and $U _ { 0 }$.
We introduce the following matrices $Q = \left( \begin{array} { c c } 7 & 3 \\ - 5 & 5 \end{array} \right)$ and $D = \left( \begin{array} { c c } - 0,25 & 0 \\ 0 & 1 \end{array} \right)$. a. We admit that the matrix $Q$ is invertible and that $Q ^ { - 1 } = \left( \begin{array} { c c } 0,1 & - 0,06 \\ 0,1 & 0,14 \end{array} \right)$. Show that $Q \times D \times Q ^ { - 1 } = A$. b. Show by induction on $n$ that for any non-zero natural integer $n$: $A ^ { n } = Q \times D ^ { n } \times Q ^ { - 1 }$. c. For any non-zero natural integer $n$, determine $D ^ { n }$ as a function of $n$.
We admit that for any non-zero natural integer $n$, $$A ^ { n } = \left( \begin{array} { l l }
0,3 + 0,7 \times ( - 0,25 ) ^ { n } & 0,42 - 0,42 \times ( - 0,25 ) ^ { n } \\
0,5 - 0,5 \times ( - 0,25 ) ^ { n } & 0,7 + 0,3 \times ( - 0,25 ) ^ { n }
\end{array} \right)$$ a. Deduce the expressions of $j _ { n }$ and $a _ { n }$ as functions of $n$ and determine the limits of these two sequences. b. What can we conclude about the population of animals studied?
We study the evolution over time of the number of young and adult animals in a population.\\
For any natural integer $n$, we denote by $j _ { n }$ the number of young animals after $n$ years of observation and $a _ { n }$ the number of adult animals after $n$ years of observation.\\
At the beginning of the first year of the study, there are 200 young animals and 500 adult animals. Thus $j _ { 0 } = 200$ and $a _ { 0 } = 500$.\\
We admit that for any natural integer $n$ we have:
$$\left\{ \begin{array} { l }
j _ { n + 1 } = 0,125 j _ { n } + 0,525 a _ { n } \\
a _ { n + 1 } = 0,625 j _ { n } + 0,625 a _ { n }
\end{array} \right.$$
We introduce the following matrices:
$$A = \left( \begin{array} { l l }
0,125 & 0,525 \\
0,625 & 0,625
\end{array} \right) \text { and, for any natural integer } n , U _ { n } = \binom { j _ { n } } { a _ { n } } .$$
\begin{enumerate}
\item a. Show that for any natural integer $n , U _ { n + 1 } = A \times U _ { n }$.\\
b. Calculate the number of young animals and adult animals after one year of observation and then after two years of observation (results rounded down to the nearest unit).\\
c. For any non-zero natural integer $n$, express $U _ { n }$ as a function of $A ^ { n }$ and $U _ { 0 }$.
\item We introduce the following matrices $Q = \left( \begin{array} { c c } 7 & 3 \\ - 5 & 5 \end{array} \right)$ and $D = \left( \begin{array} { c c } - 0,25 & 0 \\ 0 & 1 \end{array} \right)$.\\
a. We admit that the matrix $Q$ is invertible and that $Q ^ { - 1 } = \left( \begin{array} { c c } 0,1 & - 0,06 \\ 0,1 & 0,14 \end{array} \right)$.\\
Show that $Q \times D \times Q ^ { - 1 } = A$.\\
b. Show by induction on $n$ that for any non-zero natural integer $n$: $A ^ { n } = Q \times D ^ { n } \times Q ^ { - 1 }$.\\
c. For any non-zero natural integer $n$, determine $D ^ { n }$ as a function of $n$.
\item We admit that for any non-zero natural integer $n$,
$$A ^ { n } = \left( \begin{array} { l l }
0,3 + 0,7 \times ( - 0,25 ) ^ { n } & 0,42 - 0,42 \times ( - 0,25 ) ^ { n } \\
0,5 - 0,5 \times ( - 0,25 ) ^ { n } & 0,7 + 0,3 \times ( - 0,25 ) ^ { n }
\end{array} \right)$$
a. Deduce the expressions of $j _ { n }$ and $a _ { n }$ as functions of $n$ and determine the limits of these two sequences.\\
b. What can we conclude about the population of animals studied?
\end{enumerate}