Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Problem 6: Mathematical Induction Proof**
**Statement to Prove:**
Prove by induction that for the Fibonacci sequence:
\[ 3F_n = F_{n+2} + F_{n-2} \]
for \( n \geq 2 \).
**Explanation:**
This problem involves proving a property of the Fibonacci sequence using the method of mathematical induction. The Fibonacci sequence is defined as follows:
- \( F_0 = 0 \)
- \( F_1 = 1 \)
- \( F_n = F_{n-1} + F_{n-2} \) for \( n \geq 2 \)
**Steps to Prove by Induction:**
1. **Base Case:** Verify the statement for the initial value (\( n = 2 \)).
2. **Inductive Step:** Assume the statement holds for \( n = k \), i.e., \( 3F_k = F_{k+2} + F_{k-2} \).
3. **Prove for \( n = k+1 \):** Show that if the statement is true for \( n = k \), then it must also be true for \( n = k+1 \).
This problem requires verifying these steps to complete the proof, ensuring the validity of the given formula across the domain specified (\( n \geq 2 \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F734a9f72-654e-4f0b-bd3d-c0fbcfb94ea0%2F22ea9c57-27eb-4d99-be6c-2ed152337fe0%2Fswso3zg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Mathematical Induction Proof**
**Statement to Prove:**
Prove by induction that for the Fibonacci sequence:
\[ 3F_n = F_{n+2} + F_{n-2} \]
for \( n \geq 2 \).
**Explanation:**
This problem involves proving a property of the Fibonacci sequence using the method of mathematical induction. The Fibonacci sequence is defined as follows:
- \( F_0 = 0 \)
- \( F_1 = 1 \)
- \( F_n = F_{n-1} + F_{n-2} \) for \( n \geq 2 \)
**Steps to Prove by Induction:**
1. **Base Case:** Verify the statement for the initial value (\( n = 2 \)).
2. **Inductive Step:** Assume the statement holds for \( n = k \), i.e., \( 3F_k = F_{k+2} + F_{k-2} \).
3. **Prove for \( n = k+1 \):** Show that if the statement is true for \( n = k \), then it must also be true for \( n = k+1 \).
This problem requires verifying these steps to complete the proof, ensuring the validity of the given formula across the domain specified (\( n \geq 2 \)).
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