Hi, I am struggling with the attached HW problem. Any help is appreciated. Thanks. **Mathematical Problems**1. **Proof and Problem Statements:** (i) Show that $ \sqrt{7} $ is irrational. (Proof by contradiction) **Hint:** Begin by assuming that $ \sqrt{7} $ is rational, and express it as $ \frac{p}{q} $ in lowest terms. (ii) Suppose $ 0 < a < b $. Prove that: $ a < \sqrt{ab} < b $ and $ \sqrt{ab} \leq \frac{1}{2}(a + b) $. **Hint:** Use properties of inequalities and the arithmetic mean-geometric mean inequality (AM-GM Inequality). (iii) Suppose that $ x $ and $ y $ satisfy $ \frac{x}{3} + \frac{y}{3} = 1 $. Prove that $ x^2 + y^2 > 1 $. **Hint:** Consider solving for $ x $ and $ y $ and using the method of completing the square or analyzing the expressions geometrically.These problems aim to test understanding of irrationality proofs, inequality manipulations, and algebraic problem-solving techniques.

Name: Hi, I am struggling with the attached HW problem. Any help is apprecia
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Description: Hi, I am struggling with the attached HW problem. Any help is appreciated. Thanks. **Mathematical Problems**1. **Proof and Problem Statements:** (i) Show that $ \sqrt{7} $ is irrational. (Proof by contradiction) **Hint:** Begin by assuming that

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Answered: 1. (i) Show that v7 is irrational.…

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1. (i) Show that v7 is irrational. (Proof by contradiction) (ii) Suppose 0 < a < b. Prove that: a < Vab < b and Vab< }(a+b). (iii) Suppose that x and y satisfy+ = 1. Prove that a² + y? > 1.

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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Hi,

I am struggling with the attached HW problem. Any help is appreciated. Thanks.

**Mathematical Problems**

1. **Proof and Problem Statements:**

(i) Show that $ \sqrt{7} $ is irrational. (Proof by contradiction)

**Hint:** Begin by assuming that $ \sqrt{7} $ is rational, and express it as $ \frac{p}{q} $ in lowest terms.

(ii) Suppose $ 0 < a < b $. Prove that: $ a < \sqrt{ab} < b $ and $ \sqrt{ab} \leq \frac{1}{2}(a + b) $.

**Hint:** Use properties of inequalities and the arithmetic mean-geometric mean inequality (AM-GM Inequality).

(iii) Suppose that $ x $ and $ y $ satisfy $ \frac{x}{3} + \frac{y}{3} = 1 $. Prove that $ x^2 + y^2 > 1 $.

**Hint:** Consider solving for $ x $ and $ y $ and using the method of completing the square or analyzing the expressions geometrically.

These problems aim to test understanding of irrationality proofs, inequality manipulations, and algebraic problem-solving techniques.

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