We are interested in the development of a bacterium. In this exercise, we model its development with the following assumptions: this bacterium has a probability of 0.3 of dying without offspring and a probability of 0.7 of dividing into two daughter bacteria. In the context of this experiment, we admit that the reproduction laws of bacteria are the same for all generations of bacteria whether they are mother or daughter. For any natural integer $n$, we call $p _ { n }$ the probability of obtaining at most $n$ descendants for a bacterium. We admit that, according to this model, the sequence $\left( p _ { n } \right)$ is defined as follows: $p _ { 0 } = 0.3$ and, for any natural integer $n$, $$p _ { n + 1 } = 0.3 + 0.7 p _ { n } ^ { 2 }$$
The spreadsheet below gives approximate values of the sequence $\left( p _ { n } \right)$ a. Determine the exact values of $p _ { 1 }$ and $p _ { 2 }$ (hidden in the spreadsheet) and interpret these values in the context of the problem. b. What is the probability, rounded to $10 ^ { - 3 }$, of obtaining at least 11 generations of bacteria starting from a bacterium of this type? c. Make conjectures about the variations and convergence of the sequence $\left( p _ { n } \right)$.
a. Prove by induction on $n$ that, for any natural integer $n , 0 \leqslant p _ { n } \leqslant p _ { n + 1 } \leqslant 0.5$. b. Justify that the sequence $\left( p _ { n } \right)$ is convergent.
We call $L$ the limit of the sequence $\left( p _ { n } \right)$. a. Justify that $L$ is a solution of the equation $$0.7 x ^ { 2 } - x + 0.3 = 0$$ b. Then determine the limit of the sequence $\left( p _ { n } \right)$.
A
B
1
$n$
$p _ { n }$
2
0
0,3
3
1
4
2
5
3
0,40769562
6
4
0,416351
7
5
0,42134371
8
6
0,42427137
9
7
0,42600433
10
8
0,427 03578
11
9
0,42765169
12
10
0,428 02018
13
11
0,42824089
14
12
0,42837318
15
13
0,42845251
16
14
0,42850009
17
15
0,42852863
18
16
0,42854575
19
17
0,42855602
The following function, written in Python language, aims to return the first $n$ terms of the sequence $\left( p _ { n } \right)$. \begin{verbatim} def suite(n) : p= ... s=[p] for i in range (...) : p=... s.append(p) return (s) \end{verbatim} Rewrite this function on your answer sheet, completing lines 2, 4 and 5 so that the function suite(n) returns, in the form of a list, the first $n$ terms of the sequence.
We are interested in the development of a bacterium.\\
In this exercise, we model its development with the following assumptions: this bacterium has a probability of 0.3 of dying without offspring and a probability of 0.7 of dividing into two daughter bacteria.\\
In the context of this experiment, we admit that the reproduction laws of bacteria are the same for all generations of bacteria whether they are mother or daughter.\\
For any natural integer $n$, we call $p _ { n }$ the probability of obtaining at most $n$ descendants for a bacterium.\\
We admit that, according to this model, the sequence $\left( p _ { n } \right)$ is defined as follows:\\
$p _ { 0 } = 0.3$ and, for any natural integer $n$,
$$p _ { n + 1 } = 0.3 + 0.7 p _ { n } ^ { 2 }$$
\begin{enumerate}
\item The spreadsheet below gives approximate values of the sequence $\left( p _ { n } \right)$\\
a. Determine the exact values of $p _ { 1 }$ and $p _ { 2 }$ (hidden in the spreadsheet) and interpret these values in the context of the problem.\\
b. What is the probability, rounded to $10 ^ { - 3 }$, of obtaining at least 11 generations of bacteria starting from a bacterium of this type?\\
c. Make conjectures about the variations and convergence of the sequence $\left( p _ { n } \right)$.
\item a. Prove by induction on $n$ that, for any natural integer $n , 0 \leqslant p _ { n } \leqslant p _ { n + 1 } \leqslant 0.5$.\\
b. Justify that the sequence $\left( p _ { n } \right)$ is convergent.
\item We call $L$ the limit of the sequence $\left( p _ { n } \right)$.\\
a. Justify that $L$ is a solution of the equation
$$0.7 x ^ { 2 } - x + 0.3 = 0$$
b. Then determine the limit of the sequence $\left( p _ { n } \right)$.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
& A & B \\
\hline
1 & $n$ & $p _ { n }$ \\
\hline
2 & 0 & 0,3 \\
\hline
3 & 1 & \\
\hline
4 & 2 & \\
\hline
5 & 3 & 0,40769562 \\
\hline
6 & 4 & 0,416351 \\
\hline
7 & 5 & 0,42134371 \\
\hline
8 & 6 & 0,42427137 \\
\hline
9 & 7 & 0,42600433 \\
\hline
10 & 8 & 0,427 03578 \\
\hline
11 & 9 & 0,42765169 \\
\hline
12 & 10 & 0,428 02018 \\
\hline
13 & 11 & 0,42824089 \\
\hline
14 & 12 & 0,42837318 \\
\hline
15 & 13 & 0,42845251 \\
\hline
16 & 14 & 0,42850009 \\
\hline
17 & 15 & 0,42852863 \\
\hline
18 & 16 & 0,42854575 \\
\hline
19 & 17 & 0,42855602 \\
\hline
\end{tabular}
\end{center}
The following function, written in Python language, aims to return the first $n$ terms of the sequence $\left( p _ { n } \right)$.
\begin{verbatim}
def suite(n) :
p= ...
s=[p]
for i in range (...) :
p=...
s.append(p)
return (s)
\end{verbatim}
Rewrite this function on your answer sheet, completing lines 2, 4 and 5 so that the function suite(n) returns, in the form of a list, the first $n$ terms of the sequence.