How to Solve Harmonic Progression Problems
Solve harmonic progression aptitude problems by inverting to an AP, plus the harmonic mean formula, with worked examples and practice questions.
Expected Interview Answer
A harmonic progression (HP) is a sequence whose reciprocals form an arithmetic progression, so the standard technique is to invert every term, solve the resulting AP problem with the a + (n−1)d formula, then invert the answer back.
There is no direct sum formula for an HP itself — only for the reciprocal AP — which is why every HP problem is really an AP problem wearing a disguise. The harmonic mean of two numbers x and y is 2xy/(x+y), which equals the reciprocal of the average of the reciprocals 1/x and 1/y. For three numbers in HP, their reciprocals are in AP, so the middle reciprocal equals the average of the outer two reciprocals. A frequent real-world source of HP is rate problems: average speed over equal distances at different speeds is a harmonic mean, not an arithmetic one.
- Inverting to AP is the single technique that solves every HP problem
- The harmonic mean formula directly handles two-number rate averages
- Recognizing HP in disguise (like average speed) avoids a common averaging error
AI Mentor Explanation
A bowler whose overs-per-wicket figures are 2, 4, 6 doesn’t have runs or wickets in harmonic progression directly, but if you flip those numbers to wickets-per-over — 1/2, 1/4, 1/6 — you’d check whether that reciprocal sequence is arithmetic to confirm an HP. The core HP technique is always this: invert the terms, treat the result as a plain AP with a + (n−1)d, solve it, then invert back to answer the original question about the bowler’s figures.
Worked example
Invert to AP
- Reciprocals: 3, 5, 7
- a=3, d=2
4th AP term
- T4 = 3 + 3×2 = 9
Invert back
- 4th HP term = 1/9
Step-by-Step Explanation
Step 1
Invert every term
Take the reciprocal of each HP term to obtain a candidate AP.
Step 2
Solve as an AP
Find a and d for the reciprocal sequence, then apply Tn = a + (n−1)d.
Step 3
Invert the result back
Take the reciprocal of the AP answer to get the actual HP term.
Step 4
Use harmonic mean for two values
For two numbers x, y, the harmonic mean is 2xy/(x+y).
What Interviewer Expects
- Recognizing HP has no direct formula — must invert to AP first
- Correct use of the harmonic mean formula 2xy/(x+y)
- Identifying average-speed-over-equal-distance as a harmonic mean scenario
- Correctly inverting the final AP answer back to the HP domain
Common Mistakes
- Applying the AP sum formula directly to HP terms without inverting first
- Using a simple arithmetic average for rate/speed problems that need a harmonic mean
- Forgetting to invert the final answer back after solving the reciprocal AP
- Assuming a sequence is HP without verifying its reciprocals form a genuine AP
Best Answer (HR Friendly)
“A harmonic progression is any sequence whose reciprocals form a plain arithmetic progression, so the trick is always to flip every term, solve it as a normal AP problem, and then flip the answer back. The other place this shows up is averaging rates — like average speed over equal distances at different speeds — where the correct average is the harmonic mean, 2xy over x plus y, not the simple average.”
Follow-up Questions
- How do you find the harmonic mean of more than two numbers?
- Why is average speed over equal distances a harmonic mean and not an arithmetic mean?
- How does the relationship AM ≥ GM ≥ HM apply to a set of positive numbers?
- How would you insert harmonic means between two given numbers?
MCQ Practice
1. Find the harmonic mean of 4 and 6.
HM = 2×4×6/(4+6) = 48/10 = 4.8.
2. A car travels equal distances at 30km/h and 50km/h. Its average speed for the whole trip is?
HM = 2×30×50/(30+50) = 3000/80 = 37.5 km/h.
3. If 1/x, 1/y, 1/z are in AP, then x, y, z are said to be in?
By definition, a sequence is a harmonic progression exactly when its reciprocals form an AP.
Flash Cards
Definition of harmonic progression? — A sequence whose reciprocals form an arithmetic progression.
Harmonic mean of x and y? — HM = 2xy / (x + y).
Method for any HP problem? — Invert terms → solve as AP → invert the result back.
When does average speed need a harmonic mean? — When equal distances are covered at different speeds.