bac-s-maths 2018 QIII.B.4

bac-s-maths · France · centres-etrangers Geometric Sequences and Series Prove a Transformed Sequence is Geometric

For $n \geqslant 1$, we set $p_n = P(A_n)$ with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$. We set for all integer $n \geqslant 1$: $v_n = p_n - 0{,}8$. a. Prove that $(v_n)$ is a geometric sequence and give its first term $v_1$ and common ratio. b. Express $v_n$ as a function of $n$. Deduce that, for all $n \geqslant 1$, $p_n = 0{,}8 + 0{,}2 \times 0{,}5^{n-1}$. c. Determine the limit of the sequence $(p_n)$.

For $n \geqslant 1$, we set $p_n = P(A_n)$ with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$. We set for all integer $n \geqslant 1$: $v_n = p_n - 0{,}8$.\\
a. Prove that $(v_n)$ is a geometric sequence and give its first term $v_1$ and common ratio.\\
b. Express $v_n$ as a function of $n$. Deduce that, for all $n \geqslant 1$, $p_n = 0{,}8 + 0{,}2 \times 0{,}5^{n-1}$.\\
c. Determine the limit of the sequence $(p_n)$.

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