Define (xn)n≥1 by x1 := 1 and xn+1 := (3xn + 3)/4 for n ≥ 1. Prove thatlimn→∞ xn = 3.(Hint: First show that xn ≤ 3 for all n ≥ 1.)

Question

Define (xn)n≥1 by x1 := 1 and xn+1 := (3xn + 3)/4 for n ≥ 1. Prove thatlimn→∞ xn = 3.(Hint: First show that xn ≤ 3 for all n ≥ 1.)
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Solution 1

Sure, here is the step-by-step proof:

Step 1: Prove that xn ≤ 3 for all n ≥ 1.

We will use induction to prove this.

Base case: For n=1, x1 = 1 which is less than or equal to 3.

Inductive step: Assume that xn ≤ 3 for some n ≥ 1. We need to show that xn+1 ≤ 3.

xn+1 = (3xn + 3)/4 ≤ (3*3 + 3)/4 = Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob

Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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