Use induction to verify the candidate solution for the following recurrence relation: 1 for n > 1 n(n + 1)' T(n) = T(n-1) + 1 T(1) 2 with canidate solution T(n) = 1 =
Use induction to verify the candidate solution for the following recurrence relation: 1 for n > 1 n(n + 1)' T(n) = T(n-1) + 1 T(1) 2 with canidate solution T(n) = 1 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Verifying Candidate Solutions Using Induction**
**Objective:**
Use induction to verify the candidate solution for the following recurrence relation:
\[ T(n) = T(n-1) + \frac{1}{n(n+1)} \quad \text{for } n > 1 \]
**Base Case:**
\[ T(1) = \frac{1}{2} \]
**Candidate Solution:**
The candidate solution is proposed as:
\[ T(n) = \frac{n}{n+1} \]
**Instructions:**
1. Verify the base case by substituting \( n = 1 \) into the candidate solution.
2. Use mathematical induction to prove that the solution holds for all \( n > 1 \).
3. Demonstrate the step-by-step inductive proof and ensure the calculations align with the recurrence relation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53389f30-4046-4262-ba29-305203f0e182%2Fd011f6a6-e182-488c-bdc8-c25e16e8b85d%2F8qeflb_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Verifying Candidate Solutions Using Induction**
**Objective:**
Use induction to verify the candidate solution for the following recurrence relation:
\[ T(n) = T(n-1) + \frac{1}{n(n+1)} \quad \text{for } n > 1 \]
**Base Case:**
\[ T(1) = \frac{1}{2} \]
**Candidate Solution:**
The candidate solution is proposed as:
\[ T(n) = \frac{n}{n+1} \]
**Instructions:**
1. Verify the base case by substituting \( n = 1 \) into the candidate solution.
2. Use mathematical induction to prove that the solution holds for all \( n > 1 \).
3. Demonstrate the step-by-step inductive proof and ensure the calculations align with the recurrence relation.
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