H(n) = 2H(n − 1) + 1, n ≥ 2, H(1) = 1 ⇒ H(n) = 2" – 1↓ Using mathematical induction, verify that our closed form solution is correct.
H(n) = 2H(n − 1) + 1, n ≥ 2, H(1) = 1 ⇒ H(n) = 2" – 1↓ Using mathematical induction, verify that our closed form solution is correct.
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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![**Recursive Formula and Closed Form Solution Verification**
Given the recursive equation:
\[ H(n) = 2H(n-1) + 1, \quad n \geq 2, \quad H(1) = 1 \]
we need to verify the closed form solution:
\[ H(n) = 2^n - 1 \]
**Task:**
Using mathematical induction, verify that our closed form solution is correct.
**Explanation:**
- **Base Case:** Check if the closed form is true for \( n = 1 \).
- **Inductive Step:** Assume it holds for some \( n = k \), that is, \( H(k) = 2^k - 1 \). Show it holds for \( n = k+1 \).
- Substitute \( H(k+1) = 2H(k) + 1 \) and use the induction hypothesis to find \( H(k+1) \).
This process ensures that the closed form solution matches the recursive formulation for all \( n \geq 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd61d70a8-1828-415d-bba3-28735268e091%2F98fcda10-fb32-4d0a-b8e6-3dbf82421121%2Fkma76ao_processed.png&w=3840&q=75)
Transcribed Image Text:**Recursive Formula and Closed Form Solution Verification**
Given the recursive equation:
\[ H(n) = 2H(n-1) + 1, \quad n \geq 2, \quad H(1) = 1 \]
we need to verify the closed form solution:
\[ H(n) = 2^n - 1 \]
**Task:**
Using mathematical induction, verify that our closed form solution is correct.
**Explanation:**
- **Base Case:** Check if the closed form is true for \( n = 1 \).
- **Inductive Step:** Assume it holds for some \( n = k \), that is, \( H(k) = 2^k - 1 \). Show it holds for \( n = k+1 \).
- Substitute \( H(k+1) = 2H(k) + 1 \) and use the induction hypothesis to find \( H(k+1) \).
This process ensures that the closed form solution matches the recursive formulation for all \( n \geq 1 \).
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