Math Problem Statement

The exercise involves a recursive sequence (u_n) defined by u_0 = 1 and the recurrence relation u_(n+1) = u_n / (3 - 4u_n). The problem includes questions to prove properties of the sequence, deduce its monotonicity, express its terms explicitly, and analyze convergence.

Solution

The exercise is about a recursive sequence (un)(u_n), with the first term u0=1u_0 = 1, and the recursive relation given by:
un+1=un34un.u_{n+1} = \frac{u_n}{3 - 4u_n}.

The questions in the exercise are:


  1. Question 1: Show that the sequence (un)(u_n) is well-defined and un<0u_n < 0 for all nNn \in \mathbb{N}.

  2. Question 2: Based on this, deduce whether the sequence (un)(u_n) is increasing or decreasing and justify.

  3. Question 3: (a) Define a new sequence vnv_n such that vn=1unv_n = \frac{-1}{u_n}.
    (b) Prove that (vn)(v_n) is a geometric sequence, and find its common ratio qq and initial value v0v_0.

  4. Question 4: (a) Express unu_n in terms of nn.
    (b) Deduce whether the sequence (un)(u_n) converges, and if so, find its limit.


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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Recursive Sequences
Convergence of Sequences
Geometric Sequences

Formulas

Recursive relation: u_(n+1) = u_n / (3 - 4u_n)
Transformation: v_n = -1/u_n
Geometric sequence general formula: v_n = v_0 * q^n

Theorems

Convergence of bounded and monotonic sequences
Properties of geometric sequences

Suitable Grade Level

Undergraduate (University Level) or Advanced High School (Grades 11-12)