Continue with Geometric Series

• Importance: Common in mocks, prone to errors.
  • Stats: Only 10-12% get these questions right in mocks.

Key Concept: Triangle Inequality Theorem

• Theorem Statement: For any triangle, the sum of any two sides must be greater than the third side.
  • Example: If sides are labeled p, q, and r, then:
    • p + q > r
    • p + r > q
    • q + r > p

Problem-Solving Strategy

1. Given Condition: Sum of two sides is greater than the third side.
2. Equation: p² + q² >= 2pq
3. Transfer to form: (p - q)² >= 0
   • Always true because a perfect square is non-negative.
4. New Formulation:
   • Add and manipulate sides, example:
     • p² + q² greater than another term.
   • When transforming, ensure addition aligns with the inequality:
     • (p + q)² >= 2pq
5. Insight: p² + q² is always positive.
6. Finding Correct Answer:
   • Simplify to fit into solution space.
   • Example simplification: p² + q² >= r²

Common Pitfalls

• Missteps: Incorrectly introducing negative signs can lead to confusion.
• Tip: Always ensure transformations preserve positive squares.

Application and Solution

1. Inequalities and Relationships:
   • p² + q² >= 2pq is greater than 0
   • Leads to solutions fitting the criteria outlined.
2. Final Solution:
   • Identify correct transformation to arrive at the right answer.
   • Example: p² + q²/r² >= 1/2 leads to solution.
     • Answer: 1/2

Practical Use

• Triangular Relations: Commonly used in various geometric problems.
• Reminder: Implement these methods to approach complex mocks and real-world geometric scenarios effectively.