Harmonic Progression (HP): $a_1$, $a_2$, ..., $a_n$ is HP iff 1/$a_1$, 1/$a_2$, ..., 1/$a_n$ is AP.

There is NO direct formula for sum of HP. Convert to AP, work with reciprocals.

Harmonic Mean (HM):

• HM of a, b = 2$\frac{ab}{a+b}$
• HM of $a_1$, ..., $a_n$ = n / (1/$a_1$ + 1/$a_2$ + ... + 1/$a_n$)

AM-GM-HM Inequality (for positive reals): AM >= GM >= HM, with equality iff all numbers are equal.

For two positive numbers a, b:

• AM = $\frac{a+b}{2}$
• GM = sqrt(ab)
• HM = 2$\frac{ab}{a+b}$
• AM * HM = $GM^2$ (always!)

JEE application: To prove x + 1/x >= 2 for x > 0, apply AM-GM: $\frac{x + 1/x}{2}$ >= sqrt$\frac{x * 1}{x}$ = 1.