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
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
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