(a) The mathematical proof of the statement stating that if { a n } be the sequence defined recursively by a 0 = 0 , a n + 1 = 2 + a n and if a n < 2 , then a n + 1 < 2 and also the conclusion by induction that a n < 2 for all n.

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Answer 1

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Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

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Answer 2

Question

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

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Answer 3

Question

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

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Answer 4

Question

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

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Answer 5

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

Answer 6

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

Answer 7

Chapter 11.1, Problem 84E

To determine

(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

Answer 8

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

Answer 9

To determine

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

Answer 10

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

Answer 11

To determine

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

Answer 12

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Answer 13

Expert Solution & Answer

Answer 14

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Answer 15

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

(a) The mathematical proof of the statement stating that if { a n } be the sequence defined recursively by a 0 = 0 , a n + 1 = 2 + a n and if a n < 2 , then a n + 1 < 2 and also the conclusion by induction that a n < 2 for all n.

Solution Summary: The author explains the mathematical proof of the statement stating that if a n be the sequence defined recursively by 0 = 0, i.e.

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(a)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n + 1} < 2$ and also the conclusion by induction that $a_{n} < 2$ for all n.

(b)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and if $a_{n} < 2$ , then $a_{n} \leq a_{n + 1}$ and also the conclusion by induction that ${a_{n}}$ is increasing.

(c)

The mathematical proof of the statement stating that if ${a_{n}}$ be the sequence defined recursively by $a_{0} = 0, a_{n + 1} = \sqrt{2 + a_{n}}$ and by using (a) and (b), L= $\lim_{n \to \infty} a_{n}$ exists and $L = \sqrt{2 + L}$ along with value of L.

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Chapter 11 Solutions

(a) The mathematical proof of the statement stating that if { a n } be the sequence defined recursively by a 0 = 0 , a n + 1 = 2 + a n and if a n &lt; 2 , then a n + 1 &lt; 2 and also the conclusion by induction that a n &lt; 2 for all n.

Solution Summary: The author explains the mathematical proof of the statement stating that if a n be the sequence defined recursively by 0 = 0, i.e.

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