bac-s-maths 2018 QIII.B.3

bac-s-maths · France · centres-etrangers Sequences and series, recurrence and convergence Convergence proof and limit determination

For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$'', with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$ for all $n \geqslant 1$. a. Prove by induction that, for all integer $n \geqslant 1$: $p_n > 0{,}8$. b. Prove that the sequence $(p_n)$ is decreasing. c. Is the sequence $(p_n)$ convergent?

For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$'', with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$ for all $n \geqslant 1$.\\
a. Prove by induction that, for all integer $n \geqslant 1$: $p_n > 0{,}8$.\\
b. Prove that the sequence $(p_n)$ is decreasing.\\
c. Is the sequence $(p_n)$ convergent?

Paper Questions

QI.1 QI.2 QI.3 QII.1 QII.2 QII.3 QII.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIV.A QIV.B QIV.C