Harmonic progression
- Olympiad · IOQM
A harmonic progression is a list of numbers whose flips, 1 over each number, go up or down by the same step, like 1, 1/2, 1/3, 1/4.
The reciprocal of a number x is 1/x. Take 1, 1/2, 1/3, 1/4. Their reciprocals are 1, 2, 3, 4. These go up by 1 each time, so they form an arithmetic progression. That makes 1, 1/2, 1/3, 1/4 a harmonic progression, or H.P.
There is no simple formula for the sum of an H.P. So most work on it means turning it into an A.P., doing the work there, then turning back.
A small example
Find the next term of 1/3, 1/5, 1/7.
- Flip each term: 3, 5, 7. This is an A.P. with common difference 2.
- The next term of the A.P. is 9.
- Flip back: the next term of the H.P. is 1/9.
The harmonic mean of a and b is 2ab/(a + b). It shows up in average speed. You ride 60 km at 30 km/h and come back at 20 km/h. The trip out takes 2 hours and the trip back takes 3. So you cover 120 km in 5 hours, which is 24 km/h. The harmonic mean gives the same: 2 × 30 × 20 / 50 = 24. The plain average, 25, is wrong here.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under algebra. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Algebra) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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