At the beginning of the experiment, we have a piece of 2 g of polonium. We know that 1 g of polonium contains $3 \times 10^{21}$ atomic nuclei. We assume that, after 24 hours, $0.5\%$ of the nuclei have disintegrated and that, to compensate for this loss, we then add $0.005\text{ g}$ of polonium. We model the situation using a sequence $\left(v_n\right)_{n \in \mathbb{N}}$; we denote $v_0$ the number of nuclei contained in the polonium at the beginning of the experiment. For $n \geqslant 1$, $v_n$ denotes the number of nuclei contained in the polonium after $n$ days have elapsed.
a. Verify that $v_0 = 6 \times 10^{21}$. b. Explain that, for every natural number $n$, we have $$v_{n+1} = 0{,}995\, v_n + 1{,}5 \times 10^{19}.$$
a. Prove, by induction on $n$, that $0 \leqslant v_{n+1} \leqslant v_n$. b. Deduce that the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ is convergent.
We consider the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ defined, for every natural number $n$, by: $$u_n = v_n - 3 \times 10^{21}.$$ a. Show that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is geometric with common ratio 0.995. b. Deduce that, for every natural number $n$, $v_n = 3 \times 10^{21}\left(0{,}995^n + 1\right)$. c. Deduce the limit of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ and interpret the result in the context of the exercise.
Determine, by calculation, after how many days the number of polonium nuclei will be less than $4{,}5 \times 10^{21}$. Justify the answer.
We wish to have the list of terms of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$. For this, we use a function called \texttt{noyaux} programmed in Python language and partially transcribed below. \begin{verbatim} def noyaux (n) : V =6*10**21 L=[V] for k in range (n) : V=... L.append(V) return L \end{verbatim} a. From reading the previous questions, propose two different solutions to complete line 5 of the \texttt{noyaux} function so that it answers the problem. b. For which value of the integer $n$ will the command \texttt{noyaux(n)} return the daily records of the number of nuclei contained in the polonium sample during 52 weeks of study?
At the beginning of the experiment, we have a piece of 2 g of polonium. We know that 1 g of polonium contains $3 \times 10^{21}$ atomic nuclei. We assume that, after 24 hours, $0.5\%$ of the nuclei have disintegrated and that, to compensate for this loss, we then add $0.005\text{ g}$ of polonium. We model the situation using a sequence $\left(v_n\right)_{n \in \mathbb{N}}$; we denote $v_0$ the number of nuclei contained in the polonium at the beginning of the experiment. For $n \geqslant 1$, $v_n$ denotes the number of nuclei contained in the polonium after $n$ days have elapsed.
\begin{enumerate}
\item a. Verify that $v_0 = 6 \times 10^{21}$.\\
b. Explain that, for every natural number $n$, we have
$$v_{n+1} = 0{,}995\, v_n + 1{,}5 \times 10^{19}.$$
\item a. Prove, by induction on $n$, that $0 \leqslant v_{n+1} \leqslant v_n$.\\
b. Deduce that the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ is convergent.
\item We consider the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ defined, for every natural number $n$, by:
$$u_n = v_n - 3 \times 10^{21}.$$
a. Show that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is geometric with common ratio 0.995.\\
b. Deduce that, for every natural number $n$, $v_n = 3 \times 10^{21}\left(0{,}995^n + 1\right)$.\\
c. Deduce the limit of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ and interpret the result in the context of the exercise.
\item Determine, by calculation, after how many days the number of polonium nuclei will be less than $4{,}5 \times 10^{21}$. Justify the answer.
\item We wish to have the list of terms of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$. For this, we use a function called \texttt{noyaux} programmed in Python language and partially transcribed below.
\begin{verbatim}
def noyaux (n) :
V =6*10**21
L=[V]
for k in range (n) :
V=...
L.append(V)
return L
\end{verbatim}
a. From reading the previous questions, propose two different solutions to complete line 5 of the \texttt{noyaux} function so that it answers the problem.\\
b. For which value of the integer $n$ will the command \texttt{noyaux(n)} return the daily records of the number of nuclei contained in the polonium sample during 52 weeks of study?
\end{enumerate}