Module 2 · Algebra
Chapter 26: Harmonic Progression
Algebra
2 topics
What this chapter covers
The 2 topics below make up Harmonic Progression on the CAT. Click any topic to open its notes, examples and practice test.
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Formula Reference Sheet
This chapter
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 |
CAT reference
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