Harmonic Progression and Advanced Applications

Harmonic Progression and Advanced Applications

This lesson covers the following key concepts:

• Harmonic Progression (HP) definition
• nth term of HP
• Sum of HP (not standard, use reciprocals)
• Relationship with AP
• Insertion of harmonic means
• Applications of HP in problems
• Mixed problems involving AP, GP, HP

Important Formulas

• $$\text{HP: if } \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text{ are in AP}$$
• $$\text{nth term of HP: } \frac{1}{a + (n-1)d} \text{ where reciprocals form AP}$$
• $$\text{Three terms in HP: } \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \Rightarrow \frac{2}{b} = \frac{1}{a} + \frac{1}{c}$$
• $$\text{Insert m HMs between a, b: find d for AP of reciprocals}$$
• $$\text{If a, b, c in HP: } b = \frac{2ac}{a + c}$$
• $$\text{AM} \times \text{HM} = \text{GM}^2$$