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.

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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 \( a > -1 \).
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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