1. Use mathematical induction or strong mathematical induction to prove the given statement. (a). (2.n!)² < (2n)! · (n + 1), for all integers n ≥ 2.
1. Use mathematical induction or strong mathematical induction to prove the given statement. (a). (2.n!)² < (2n)! · (n + 1), for all integers n ≥ 2.
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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Discrete Math please answer both a and b
![### Induction Problems
#### 1. Use mathematical induction or strong mathematical induction to prove the given statements:
### (a)
\[
(2^n \cdot n!)^2 < (2n)! \cdot (n + 1), \text{ for all integers } n \ge 2.
\]
### (b)
Let \(\{F_n\}_{n=0}^{\infty}\) be the Fibonacci sequence. Show that \(\forall n \in \mathbb{Z}^+, F_n < \left(\frac{13}{8}\right)^n\). (Note, \(F_0 = \left(\frac{13}{8}\right)^0\))
### Explanation:
**Part (a):**
- The problem statement requires proving the inequality \((2^n \cdot n!)^2 < (2n)! \cdot (n + 1)\) using mathematical induction or strong mathematical induction for \(n \ge 2\).
**Part (b):**
- In this part, the goal is to show that for all positive integers \(n\), the \(n^{th}\) Fibonacci number \(F_n\) is less than \(\left(\frac{13}{8}\right)^n\).
- The Fibonacci sequence is defined by:
\[
F_0 = 0, \quad F_1 = 1, \quad \text{and} \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \ge 2.
\]
Both parts involve demonstrating the given inequalities using the principles of mathematical induction, a powerful method for proving assertions about positive integers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b2a143d-2016-460a-a4a8-767b87cc7d00%2Ff00fb9d6-8859-46a6-a22d-072ea448c640%2F7907itr_processed.png&w=3840&q=75)
Transcribed Image Text:### Induction Problems
#### 1. Use mathematical induction or strong mathematical induction to prove the given statements:
### (a)
\[
(2^n \cdot n!)^2 < (2n)! \cdot (n + 1), \text{ for all integers } n \ge 2.
\]
### (b)
Let \(\{F_n\}_{n=0}^{\infty}\) be the Fibonacci sequence. Show that \(\forall n \in \mathbb{Z}^+, F_n < \left(\frac{13}{8}\right)^n\). (Note, \(F_0 = \left(\frac{13}{8}\right)^0\))
### Explanation:
**Part (a):**
- The problem statement requires proving the inequality \((2^n \cdot n!)^2 < (2n)! \cdot (n + 1)\) using mathematical induction or strong mathematical induction for \(n \ge 2\).
**Part (b):**
- In this part, the goal is to show that for all positive integers \(n\), the \(n^{th}\) Fibonacci number \(F_n\) is less than \(\left(\frac{13}{8}\right)^n\).
- The Fibonacci sequence is defined by:
\[
F_0 = 0, \quad F_1 = 1, \quad \text{and} \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \ge 2.
\]
Both parts involve demonstrating the given inequalities using the principles of mathematical induction, a powerful method for proving assertions about positive integers.
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