CAT Quant · Study & Practice

Harmonic Progression

AreaAlgebra DifficultyModerate CAT weightage0–1 question directly, but the AM≥GM≥HM idea recurs across Algebra, Maxima–Minima and Geometry

Harmonic Progression (HP) is the third classical progression, sitting alongside Arithmetic (AP) and Geometric (GP) progressions, and CAT loves it precisely because most students under-prepare it. The whole topic rests on one clean idea: a sequence is in HP exactly when its reciprocals form an AP. So you never solve an HP problem directly — you flip every term, work in the familiar AP world, then flip back. That single move handles nth terms, harmonic means and "find the missing term" questions. The second half of this chapter is where the real CAT marks live: the inequality AM ≥ GM ≥ HM for positive numbers, with equality only when every term is identical. This relationship quietly powers a large family of maxima–minima problems, "minimum value of x + 1/x" classics, and shortcut estimates in time–speed–distance (average speed over equal distances is the HM of the speeds). For two numbers it tightens into the elegant GM² = AM × HM, meaning the GM is the geometric mean of the other two means. This chapter builds both skills — the reciprocal trick and the mean inequality — with worked CAT-style examples, the fastest exam methods, and the traps that quietly cost careless students marks.

Topics

1 HP Basics 4 worked · 5 practice · 8-Q test Start → 2 AM-GM-HM Relationship 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Never solve an HP directly — flip every term to get an AP, solve there, then flip back.
  • HP of two numbers: HM = 2ab/(a+b). For three terms in HP the middle one is the HM of its neighbours.
  • Equal distances at speeds u and v ⇒ average speed = HM = 2uv/(u+v), never the simple average.
  • AM ≥ GM ≥ HM for positive reals; equality holds only when every term is equal — use this to read off minimums.
  • For two numbers, GM² = AM × HM, so any one mean follows instantly from the other two.
  • A sum of positive terms with a CONSTANT product is minimised when the terms are equal (drop to GM): x + k/x ≥ 2√k.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Trying to apply AP/GP nth-term formulas directly to the HP instead of to its reciprocal AP.
  • Taking the simple average of two speeds over equal distances instead of the harmonic mean.
  • Using AM ≥ GM ≥ HM on terms that can be negative or zero — it only holds for positive reals.
  • Reading the order wrong: AM is the largest and HM the smallest, not the reverse.
  • Forgetting to reciprocate the final AP answer back into an HP term.

📈 CAT exam insight & PYQ analysis

Harmonic Progression rarely appears as a standalone question in CAT, but the AM–GM–HM inequality is a recurring tool across Algebra and Maxima–Minima, where "find the minimum value of x + k/x" style problems show up almost every cycle. XAT and SNAP are more willing to ask HP nth-term and harmonic-mean questions directly. The highest-value patterns are: average speed over equal distances (HM), the GM² = AM × HM link between the three means, and minimising a sum whose terms have a fixed product. Prioritise the reciprocal trick and the equality condition — that pair unlocks most of what CAT actually tests here.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

When is a sequence in HP?Tap to reveal
When its reciprocals form an AP.
nth term of an HP?Tap to reveal
1 / (nth term of the reciprocal AP) = 1/[a+(n−1)d].
Harmonic mean of a and b?Tap to reveal
2ab/(a+b)
HM of n numbers?Tap to reveal
n / (sum of reciprocals)
The mean inequality?Tap to reveal
AM ≥ GM ≥ HM, equality iff all terms equal.
GM² in terms of the other means (two numbers)?Tap to reveal
GM² = AM × HM
Minimum of x + 1/x for x > 0?Tap to reveal
2, at x = 1
Average speed over two equal distances at u and v?Tap to reveal
2uv/(u+v) (the HM)
Which mean is largest? Smallest?Tap to reveal
AM largest, HM smallest.
Minimum of x + k/x for x > 0, k > 0?Tap to reveal
2√k
If AM = 10 and GM = 8, what is HM?Tap to reveal
6.4 (since 8² = 10 × HM)
(a+b)(1/a + 1/b) is at least?Tap to reveal
4, with equality at a = b.

📌 Quick revision

A sequence is in HP exactly when its reciprocals are in AP, so flip to an AP, solve, then flip back; the nth HP term is 1/[a+(n−1)d]. The harmonic mean of a and b is 2ab/(a+b), and average speed over equal distances is this HM. For positive reals, AM ≥ GM ≥ HM with equality only when all terms are equal, and for two numbers GM² = AM × HM. Use AM ≥ GM to minimise sums with a fixed product — x + 1/x ≥ 2, x + k/x ≥ 2√k. Traps: never average speeds simply, never apply the inequality to non-positive terms, and remember AM is largest, HM smallest.

Chapter test

📝 Harmonic Progression — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Harmonic Progression when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (2 topics)2/2
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards