Exercise 4 - Candidates who have followed the specialization course
For each of the five following propositions, indicate whether it is true or false and justify the answer chosen. One point is awarded for each correct answer correctly justified. An unjustified answer is not taken into account. An absence of answer is not penalized.
Proposition 1:
For every natural integer $n$, the units digit of $n ^ { 2 } + n$ is never equal to 4.
Proposition 2:
We consider the sequence $u$ defined, for $n \geqslant 1$, by $$u _ { n } = \frac { 1 } { n } \operatorname { gcd } ( 20 ; n ) .$$ The sequence $\left( u _ { n } \right)$ is convergent.
Proposition 3:
For all square matrices $A$ and $B$ of dimension 2, we have $A \times B = B \times A$.
Proposition 4:
A mobile can occupy two positions $A$ and $B$. At each step, it can either remain in the position it is in or change it. For every natural integer $n$, we denote $a_n$ (resp. $b_n$) the probability that the mobile is in position $A$ (resp. $B$) at step $n$.
\section*{Exercise 4 - Candidates who have followed the specialization course}
For each of the five following propositions, indicate whether it is true or false and justify the answer chosen.\\
One point is awarded for each correct answer correctly justified. An unjustified answer is not taken into account. An absence of answer is not penalized.
\section*{Proposition 1:}
For every natural integer $n$, the units digit of $n ^ { 2 } + n$ is never equal to 4.
\section*{Proposition 2:}
We consider the sequence $u$ defined, for $n \geqslant 1$, by
$$u _ { n } = \frac { 1 } { n } \operatorname { gcd } ( 20 ; n ) .$$
The sequence $\left( u _ { n } \right)$ is convergent.
\section*{Proposition 3:}
For all square matrices $A$ and $B$ of dimension 2, we have $A \times B = B \times A$.
\section*{Proposition 4:}
A mobile can occupy two positions $A$ and $B$. At each step, it can either remain in the position it is in or change it.\\
For every natural integer $n$, we denote $a_n$ (resp. $b_n$) the probability that the mobile is in position $A$ (resp. $B$) at step $n$.