Cook Salinas
07/02/2023 · Elementary School
2.14 On considère la suite \( \left(u_{n}\right) \) définie par . \( \left\{\begin{array}{l}u_{0}=0 \\ u_{n+1}=\frac{2 u_{n}+3}{u_{n}+4}\end{array}\right. \) 1. On pose, pour tout entier naturel \( n, v_{n}=\frac{u_{n}-1}{u_{n}+3} \). Montrer que la suite \( \left(v_{n}\right) \) est une suite géométrique dont on donnera le premier terme et la raison. 2. Exprimer alors \( v_{n} \), puis \( u_{n} \) en fonction de \( n \) entier naturel. 3. Déterminer la limite de la suite \( \left(v_{n}\right) \) et enfin celle de la suite \( \left(u_{n}\right) \).
UpStudy ThothAI Solution
Tutor-Verified Answer
Quick Answer
La suite \( \left(v_{n}\right) \) est géométrique avec \( v_{0} = -\frac{1}{3} \) et raison \( \frac{1}{5} \). \( v_{n} = -\frac{1}{3} \left(\frac{1}{5}\right)^{n} \) et \( u_{n} = \frac{-1 + \left(\frac{1}{5}\right)^{n}}{1 + \frac{1}{3} \left(\frac{1}{5}\right)^{n}} \). Limite de \( \left(v_{n}\right) \) : \( 0 \), limite de \( \left(u_{n}\right) \) : \( -1 \).
Step-by-step Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit