(a) The mathematical proof of the relation a n ≤ b n for all n if a n + 1 = a n b n , b n + 1 = a n + b n 2 given that the positive numbers a 1 < b 1

Question

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Answer 1

Question

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

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Answer 2

Question

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Expert Solution & Answer

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See solution

Answer 4

Question

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Expert Solution & Answer

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See solution

Answer 5

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 6

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

Answer 7

Chapter 10.1, Problem 87E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

Answer 8

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

Answer 9

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

Answer 10

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

Answer 11

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

Answer 12

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

Answer 13

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

Answer 14

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 15

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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See solution

Answer 19

Ch. 10.1 - Prob. 1PQ Ch. 10.1 - Prob. 2PQ Ch. 10.1 - Prob. 3PQ Ch. 10.1 - Prob. 4PQ Ch. 10.1 - Prob. 5PQ Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E

(a) The mathematical proof of the relation a n ≤ b n for all n if a n + 1 = a n b n , b n + 1 = a n + b n 2 given that the positive numbers a 1 &lt; b 1