HP Basics
A Harmonic Progression is any sequence whose reciprocals form an Arithmetic Progression. So 1/2, 1/5, 1/8, 1/11 … is an HP because the reciprocals 2, 5, 8, 11 … are in AP with common difference 3. The golden rule for every HP question is: never work with the HP directly — take reciprocals, solve the resulting AP, then reciprocate the answer. To find the nth term, write the reciprocal AP as a + (n−1)d, then the HP term is its reciprocal: Tₙ = 1/[a + (n−1)d]. The harmonic mean of two numbers a and b is 2ab/(a+b); if three numbers a, b, c are in HP then b is the HM of a and c. A CAT-favourite application: when equal distances are covered at speeds u and v, the average speed is exactly the HM, 2uv/(u+v) — not the simple average. Watch the sign of the AP common difference; an HP of positive decreasing terms has an increasing AP of reciprocals.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Harmonic Progression
| HP definition | a, b, c, … in HP ⇔ 1/a, 1/b, 1/c, … in AP |
|---|---|
| nth term of HP | Tₙ = 1 / [a + (n−1)d], where 1/(first term)=a, d=AP common difference |
| Harmonic Mean of two numbers | HM(a,b) = 2ab / (a + b) |
| HM of n numbers | HM = n / (1/a₁ + 1/a₂ + … + 1/aₙ) |
| Three terms in HP | b is HM of a and c ⇒ b = 2ac/(a+c) |
AM–GM–HM relationship
| The mean inequality | AM ≥ GM ≥ HM (positive reals); equality ⇔ all terms equal |
|---|---|
| AM, GM, HM of two numbers | AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b) |
| GM as the link | GM² = AM × HM (for two numbers) |
| Classic minimum | x + 1/x ≥ 2 for x>0 (AM≥GM), equality at x=1 |
| Sum–reciprocal bound | (a+b)(1/a + 1/b) ≥ 4 for a,b>0 |