Problem 22 (a) If $c>1$, show that \(c... [FREE SOLUTION]

91影视

Introduction to Real Analysis

Robert G. Bartle, Donald R. Sherbert

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 2: Problem 22

(a) If $c>1$, show that $c^{n} \geq c$ for all $n \in \mathbb{N}$, and that $c^{n}>c$ for $n>1$. (b) If $01$.

Short Answer

Expert verified

Hence, all the four statements hold true: (a) if $c>1$, then $c^{n} \geq c$ for all $n \in \mathbb{N}$ and $c^{n}>c$ for $n>1$, and (b) if $01$. This is because, when $c>1$, as we raise the power of $c$, the value increases, and when $0<c<1$, as we raise the power of $c$, the value decreases.

Step by step solution

Consider the case when $c>1$

We start with any $c>1$ and need to prove that $c^{n} \geq c$ for all $n \in \mathbb{N}$ and $c^{n}>c$ for $n>1$. Now for any $n \in \mathbb{N}$, since $c>1$, $c^{n}$ will also be greater than 1. Also, since $n \geq 1$, we can say that $c^{n} \geq c$. Now, for $n>1$, as we are increasing the power of $c$, $c^{n}$ will increase and hence $c^{n}>c$. Hence, the statement is proved.

Consider the case when \(0

We now consider any $c$ such that $01$. For such values of $c$, $c^{n}$ will always fall between 0 and 1. Also, as $n \geq 1$, we can say $c^{n} \leq c$. Now, for $n>1$, as we are increasing the power of $c$, the value of $c^{n}$ gets smaller and hence $c^{n}<c$. Hence, this statement is also proved.

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91影视

(a) If \(c>1\), show that \(c^{n} \geq c\) for all \(n \in \mathbb{N}\), and that \(c^{n}>c\) for \(n>1\). (b) If \(01\).

Short Answer

Step by step solution

Consider the case when \(c>1\)

Consider the case when \(0

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