In this exercise, we consider the sequence ( $T _ { n }$ ) defined by: $$T _ { 0 } = 180 \mathrm { and } , \text { for all natural integer } n , T _ { n + 1 } = 0,955 T _ { n } + 0,9$$
a. Prove by induction that, for all natural integer $n , T _ { n } \geqslant 20$. b. Verify that for all natural integer $n , T _ { n + 1 } - T _ { n } = - 0,045 \left( T _ { n } - 20 \right)$. Deduce the direction of variation of the sequence ( $T _ { n }$ ). c. Conclude from the above that the sequence ( $T _ { n }$ ) is convergent. Justify.
For all natural integer $n$, we set: $u _ { n } = T _ { n } - 20$. a. Show that the sequence ( $u _ { n }$ ) is a geometric sequence and specify its common ratio. b. Deduce that for all natural integer $n , T _ { n } = 20 + 160 \times 0,955 ^ { n }$. c. Calculate the limit of the sequence ( $T _ { n }$ ). d. Solve the inequality $T _ { n } \leqslant 120$ with unknown $n$ a natural integer.
In this part, we are interested in the evolution of temperature at the center of a cake after it comes out of the oven. We consider that when the cake comes out of the oven, the temperature at the center of the cake is $180 ^ { \circ } \mathrm { C }$ and that of the ambient air is $20 ^ { \circ } \mathrm { C }$. Newton's law of cooling allows us to model the temperature at the center of the cake by the previous sequence ( $T _ { n }$ ). More precisely, $T _ { n }$ represents the temperature at the center of the cake, expressed in degrees Celsius, $n$ minutes after it comes out of the oven. a. Explain why the limit of the sequence ( $T _ { n }$ ) determined in question 2. c. was foreseeable in the context of the exercise. b. We consider the following Python function:
\begin{verbatim} def temp(x) : T = 180 n = 0 while T > x : T=0.955*T+0.9 n=n+1 return n \end{verbatim} Give the result obtained by executing the command temp(120). Interpret the result in the context of the exercise.
In this exercise, we consider the sequence ( $T _ { n }$ ) defined by:
$$T _ { 0 } = 180 \mathrm { and } , \text { for all natural integer } n , T _ { n + 1 } = 0,955 T _ { n } + 0,9$$
\begin{enumerate}
\item a. Prove by induction that, for all natural integer $n , T _ { n } \geqslant 20$.\\
b. Verify that for all natural integer $n , T _ { n + 1 } - T _ { n } = - 0,045 \left( T _ { n } - 20 \right)$. Deduce the direction of variation of the sequence ( $T _ { n }$ ).\\
c. Conclude from the above that the sequence ( $T _ { n }$ ) is convergent. Justify.
\item For all natural integer $n$, we set: $u _ { n } = T _ { n } - 20$.\\
a. Show that the sequence ( $u _ { n }$ ) is a geometric sequence and specify its common ratio.\\
b. Deduce that for all natural integer $n , T _ { n } = 20 + 160 \times 0,955 ^ { n }$.\\
c. Calculate the limit of the sequence ( $T _ { n }$ ).\\
d. Solve the inequality $T _ { n } \leqslant 120$ with unknown $n$ a natural integer.
\item In this part, we are interested in the evolution of temperature at the center of a cake after it comes out of the oven.\\
We consider that when the cake comes out of the oven, the temperature at the center of the cake is $180 ^ { \circ } \mathrm { C }$ and that of the ambient air is $20 ^ { \circ } \mathrm { C }$.\\
Newton's law of cooling allows us to model the temperature at the center of the cake by the previous sequence ( $T _ { n }$ ). More precisely, $T _ { n }$ represents the temperature at the center of the cake, expressed in degrees Celsius, $n$ minutes after it comes out of the oven.\\
a. Explain why the limit of the sequence ( $T _ { n }$ ) determined in question 2. c. was foreseeable in the context of the exercise.\\
b. We consider the following Python function:
\end{enumerate}
\begin{verbatim}
def temp(x) :
T = 180
n = 0
while T > x :
T=0.955*T+0.9
n=n+1
return n
\end{verbatim}
Give the result obtained by executing the command temp(120). Interpret the result in the context of the exercise.