Exercise 6. Let a be a real number grater than -1. Prove by induction that for all positive integers n we have the following inequality: (1+ a)" 21+ na.
Exercise 6. Let a be a real number grater than -1. Prove by induction that for all positive integers n we have the following inequality: (1+ a)" 21+ na.
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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Num 6 please
![**Section 5.1 Mathematical Induction**
For each of the following exercises, follow the given steps.
(a) Determine and prove the Basis Step.
(b) For the Inductive Step, write clearly the hypothesis and the thesis.
(c) Prove the inductive step.
**Exercise 1:** Prove by induction that, for all integers \( n \geq 1 \), we have:
* [Content redacted]
**Exercise 2:** Prove by induction that, for all positive integers \( n \), we have the following inequality:
* [Content redacted]
**Exercise 3:** Prove by induction that, for all integers \( n \geq 2 \), we have:
* [Content redacted]
**Exercise 4:** Prove by induction that:
* [Content redacted]
whenever \( n \in \mathbb{Z}^+ \).
* [Content redacted]
**Exercise 6:** Let \( a \) be a real number greater than \(-1\). Prove by induction that for all positive integers \( n \) we have the following inequality:
\[
(1 + a)^n \geq 1 + na.
\]
**Note:** You need to show where you use the assumption \( a > -1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6550f6f6-7442-466d-86f7-12e7dd1263c9%2Fdb466bd2-6567-466f-8e1d-c6cf1727f072%2Ferdlesp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Section 5.1 Mathematical Induction**
For each of the following exercises, follow the given steps.
(a) Determine and prove the Basis Step.
(b) For the Inductive Step, write clearly the hypothesis and the thesis.
(c) Prove the inductive step.
**Exercise 1:** Prove by induction that, for all integers \( n \geq 1 \), we have:
* [Content redacted]
**Exercise 2:** Prove by induction that, for all positive integers \( n \), we have the following inequality:
* [Content redacted]
**Exercise 3:** Prove by induction that, for all integers \( n \geq 2 \), we have:
* [Content redacted]
**Exercise 4:** Prove by induction that:
* [Content redacted]
whenever \( n \in \mathbb{Z}^+ \).
* [Content redacted]
**Exercise 6:** Let \( a \) be a real number greater than \(-1\). Prove by induction that for all positive integers \( n \) we have the following inequality:
\[
(1 + a)^n \geq 1 + na.
\]
**Note:** You need to show where you use the assumption \( a > -1 \).
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