Math Problem Statement
Solution
The exercise is about a recursive sequence , with the first term , and the recursive relation given by:
The questions in the exercise are:
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Question 1: Show that the sequence is well-defined and for all .
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Question 2: Based on this, deduce whether the sequence is increasing or decreasing and justify.
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Question 3: (a) Define a new sequence such that .
(b) Prove that is a geometric sequence, and find its common ratio and initial value . -
Question 4: (a) Express in terms of .
(b) Deduce whether the sequence 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)