If the gambler rolls a \( 4,5,6,8,9 \), or 10 , that number is now considered the target. The dice are then rolled again and continued to be rerolled until either the sum the target, in which case the gambler wins, or the sum is a 7 in which case the gambler loses. So, if the initial dice roll had a sum of 9 and then the next sums were \( 4,4,11,3,9,12,7 \)... the gambler would win because another 9 was rolled before a 7. The probabilities that various numbers will be rolled before a 7 is as follows \[ \begin{aligned} P(4 \text { before } 7) & =\frac{3}{9} \\ P(5 \text { before } 7) & =\frac{4}{10} \\ P(6 \text { before } 7) & =\frac{5}{11} \\ P(8 \text { before } 7) & =\frac{5}{11} \\ P(9 \text { before } 7) & =\frac{4}{10} \\ P(10 \text { before } 7) & =\frac{3}{9}\end{aligned} \] So, if a gambler rolls a 6 first, the probability that they will win after that is \( \frac{5}{11} \).

Exercice 1. On considère la suite de réels définie par \( u_{0}=1 \) et pour tout \( n \in \mathbb{N}, u_{n+1}=\sqrt{2+u_{n}} \). a) Montrer que pour tout \( n \in \mathbb{N} \), on a \( 1 \leq u_{n} \leq 2 \). b) Montrer que la suite \( \left(u_{n}\right)_{n \in \mathbb{N}} \) est croissante. c) En déduire que la suite est convergente et si \( \ell \) est sa limite, justifier que \( \ell=\sqrt{2+\ell} \). d) En déduire la valeur de \( \ell \). \[ A=\{-5\} \cup\{0\} \cup\left\{u_{n}: n \geq 0\right\} . \] e) Soit Déterminer la borne supérieure et inférieure de \( A \). On justifiera.

2. \( \sqrt{51,5^{3}+51,5^{2} \cdot 26,5-51,5 \cdot 26,5^{2}-26,5^{3}} \)

For their twenty-fifth wedding anniversary, John and Abby Carter will travel to four countries. They want to go to the countries that have lots of castles, great museums, lots of mountains and where English is widely spoken. They are looking at Canada, New Zealand, England, Australia, Ireland, Switzerland and Spain. Which type of graph would best show the statistics that are being sought so they can find the four countries to travel to? Obeographical Opie Oflow

Make a bulleted list of ways interaction between Chinese society and Buddhism impacted Buddhism.

Integrate 2. \( \int_{1}^{2}(4 x-5)^{3} d x \)