AM-GM-HM and Special Series

AM-GM-HM and Special Series

This lesson covers the following key concepts:

• Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM)
• Relationship: AM ≥ GM ≥ HM
• Harmonic Progression (HP)
• Sum of first n natural numbers: Σn
• Sum of squares: Σn²
• Sum of cubes: Σn³
• Method of differences for series summation

Important Formulas

• $$\text{AM} = \frac{a+b}{2}, \text{GM} = \sqrt{ab}, \text{HM} = \frac{2ab}{a+b}$$
• $$\text{AM} \geq \text{GM} \geq \text{HM}$$
• $$\text{GM}^2 = \text{AM} \times \text{HM}$$
• $$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$
• $$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$
• $$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$$
• $$\text{HP: } \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, ...$$