Sequences / Real Analyses question

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Homework Help Overview

The discussion revolves around a problem in real analysis concerning a sequence defined by a quadratic equation. The roots of the equation are denoted as a and b, with constraints on the parameter k. The sequence is defined recursively, and participants are tasked with proving certain inequalities involving the sequence terms and determining its limit.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss determining the roots a and b in terms of k, with one participant successfully deriving these expressions. There is an exploration of proving the inequality X_{n+1} < X_n through induction, while also questioning how to establish the outer bounds a and b for the sequence. The relationship between the sequence and the roots of the quadratic equation is examined, with suggestions for a more mathematical framing of the inequalities.

Discussion Status

The discussion is active, with participants sharing insights and partial proofs. Some have made progress in proving certain inequalities, while others are still seeking clarity on how to establish the lower bound. There is a collaborative atmosphere, with participants offering suggestions and affirmations of each other's reasoning.

Contextual Notes

The problem is constrained by the requirement that 0 < k < 1/4, which influences the nature of the roots and the behavior of the sequence. Participants are also navigating the implications of the sequence's definition and its recursive nature.

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Sequences / Real Analysis question

Homework Statement


a,b are the roots of the quadratic equation x2 - x + k = 0, where 0 < k < 1/4.
(Suppose a is the smaller root). Let h belong to (a,b). The sequence xn is defined by:
x_1 = h, x_{n+1} = x^2_n + k.

Prove that a < xn+1 < xn < b, and then determine the limit of xn.

Homework Equations


The Attempt at a Solution


I have no idea how to start, if you could help me.
Thanks.
 
Last edited:
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perhaps starts by determining a and b in terms of k?
 
Okay, so I got a = \frac{1 - \sqrt{1 - 4k}}{2}, b = \frac{1 + \sqrt{1 - 4k}}{2}.

And I was able to prove X_{n+1} &lt; X_n by induction. But, I'm stuck on the outer inequalities.

EDIT: X_{n+1} &lt; X_n means that X1 = h is the largest value of Xn for all n. And h belongs to (a,b), so X1 < b, and consequently Xn < b.

I still need to prove that a is a lower bound..
 
Last edited:
I think the basic idea is:
X_{n+1} &lt; X_n \Leftrightarrow X^2_n - X_n + k &lt; 0

Therefore, Xn must be between the roots for this equation to be negative.
But is there a more mathematical way to state it?
 
Good job! How about saying x^2-x+k=(x-a)(x-b) which is negative if and only if a<x<b.
 
Oh right! Thanks a lot :)
 

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