Harmonic Progression Calculator
This harmonic progression calculator finds the nth term of an HP from its first two terms — and shows the arithmetic sequence hiding underneath every one of them, because that is where the method actually lives.
Take 1, ½, ⅓, ¼, … Turn each term upside down and you get 1, 2, 3, 4 — an ordinary arithmetic sequence stepping by 1. So the 5th term is 1 ÷ 5 = ⅕. That flip is the entire subject.
What Makes a Progression Harmonic
One thing, and it is a definition rather than a pattern you have to spot: a harmonic progression is a sequence whose reciprocals form an arithmetic progression.
So there is no separate theory to learn. Every method, every formula and every trick for harmonic progressions is an arithmetic one with a 1 ÷ wrapped around it. Flip, work in the ordinary sequence, flip back.
½, ⅕, ⅐, … flips to 2, 5, 8, stepping by 3, so the 4th term is 1 ÷ 11 = 1/11.
This calculator asks for the first two terms rather than the arithmetic difference. That is deliberate: the two terms are what you are actually looking at in the question, while the difference is a hidden quantity belonging to a sequence you have not written down yet. It works the difference out and shows it to you.
The nth Term, and Why There Is No Sum Formula
The formula every textbook quotes is:
aₙ = 1 / [a + (n − 1)d]
where a and d belong to the arithmetic progression of reciprocals, not to the harmonic one. That distinction causes more lost marks than anything else here, so it is worth reading twice. The bracket is exactly the ordinary nth-term formula; the 1 / outside it is the whole harmonic part.
Note the n − 1 as well — the same off-by-one that catches people in arithmetic sequences. Reaching the 5th term takes four steps, not five.
Now the part that is often got wrong elsewhere: there is no formula for the sum of a harmonic progression. Arithmetic progressions have one, geometric ones have one, and this family simply does not — adding reciprocals of an arithmetic sequence refuses to simplify. Any source offering a general closed form for it is mistaken. A particular sum can still be worked out exactly by adding the terms: 1 + ½ + ⅓ + ¼ + ⅕ comes to 137/60, and the sigma notation calculator will do that for any range you like.
The Term That Does Not Exist
Here is the one way harmonic progressions genuinely differ from the sequences around them, and most explanations skate over it.
Take 1/6, ¼, ½, … Flipped, that is 6, 4, 2 — stepping by −2. Carry on: the next arithmetic term is 0.
And 1 ÷ 0 is not a number. So this harmonic progression has no 4th term at all. Not infinity, not something very large — absent. The 5th term is fine again, at 1 ÷ (−2) = −½, and every term from there on carries the opposite sign.
This is why a harmonic progression can never contain 0, and why the rule matters more than it looks: a sequence with a zero in it is not an HP with an awkward term, it is a sequence with a hole where a term would be. This page marks that position and says so, rather than handing back a number that does not exist.
While we are here, one correction worth making. Some sources state that the difference must be positive for a valid harmonic progression. That is wrong: 1/6, ¼, ½ has a difference of −2 and is entirely ordinary. What actually matters is not the sign of the difference but whether the arithmetic sequence ever lands exactly on zero.
The Harmonic Mean, and the Average Speed Trap
The middle term of a three-term harmonic progression is the harmonic mean of the outer two:
HM = 2ab / (a + b)
For 3 and 6 that is (2 × 3 × 6) / 9 = 4. And here is where it earns its keep, because this is a question people get wrong constantly:
You drive somewhere at 60 km/h and come back along the same road at 40 km/h. What is your average speed?
Not 50. The answer is the harmonic mean: (2 × 60 × 40) / 100 = 48 km/h. The reason is that you spend longer at the slower speed, so it pulls the average down more than the faster one pulls it up. Whenever the same quantity is covered at different rates — equal distances, equal amounts of work — the harmonic mean is the correct average and the ordinary one is simply the wrong tool.
It is also always the smallest of the three classical averages: the arithmetic mean is largest, the geometric mean sits in the middle, and the harmonic mean is smallest, with equality only when the numbers are the same.
How to Use This Harmonic Progression Calculator
Four boxes, and you fill in any three.
1. Give the first two terms exactly as you see them — 1/6 and 1/4, say. Fractions, whole numbers and decimals such as 0.25 are all accepted, and no term may be 0.
2. Give the position or the term. Leave one box empty and it is worked out; the empty box turns dashed with a ?, then green with the answer once you press =. Fill all four instead and they are checked against each other.
3. Open the working and you get both sequences side by side: each harmonic term on top, and 1 ÷ (its arithmetic partner) underneath, so the flip is visible rather than described.
| first term a₁ | Where the progression starts. Cannot be 0 |
| second term a₂ | The next one. From these two the step is worked out |
| position n | Which term you want, counting the first as 1. A whole number |
| term aₙ | The value sitting at that position |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point. 3⁄4 stays a fraction throughout |
| ⌫ / AC | Delete one character, or clear all four boxes |
| = | Works it out |
Nothing is rounded at any stage, which matters here more than usual: harmonic terms are fractions almost by definition, and a decimal approximation of 1/7 stops being exact at the first step and drifts further with every one after.
Harmonic Progression FAQ
What is a harmonic progression?
What is the formula for the nth term of a harmonic progression?
Can a harmonic progression have a negative common difference?
Can a harmonic progression contain zero?
Is there a formula for the sum of a harmonic progression?
What is the harmonic mean of two numbers?
Flip, work, flip back — and watch for the position where the sequence underneath hits zero, because that is the only place a harmonic progression does something an arithmetic one never does.