MathDB
Convergent sequence

Source: 2017 VMO Problem 1

January 11, 2017
algebrareal analysis

Problem Statement

Given a∈Ra\in\mathbb{R} and a sequence (un)(u_n) defined by {u1=aun+1=12+2n+3n+1un+14emsp;∀n∈N∗ \begin{cases} u_1=a\\ u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}} \forall n\in\mathbb{N}^* \end{cases}
a) Prove that (un)(u_n) is convergent sequence when a=5a=5 and find the limit of the sequence in that case
b) Find all aa such that the sequence (un)(u_n) is exist and is convergent.
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