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 = 2ab/(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 = (a+b)/2
• GM = sqrt(ab)
• HM = 2ab/(a+b)
• AM * HM = GM^2 (always!)

JEE application: To prove x + 1/x >= 2 for x > 0, apply AM-GM: (x + 1/x)/2 >= sqrt(x * 1/x) = 1.