Harmonic Sequence Calculator

This calculator finds the nth term of a harmonic sequence, a list of numbers whose reciprocals form an ordinary arithmetic progression, such as 1, 1/2, 1/3, 1/4 and so on. Harmonic sequences turn up in music theory, in lens and mirror formulas in optics, in electrical resistors wired in parallel, and whenever you need the average of a set of rates like speeds or fuel efficiencies. You enter the first term and the second term of your sequence, plus the position n of the term you want to find. The calculator flips every value over to work with reciprocals, because reciprocals of a harmonic sequence always change by the same fixed amount, called the common difference. It works out that common difference, steps forward to the nth reciprocal, then flips the answer back to give you the actual nth term. Alongside the main result you also get the reciprocal (arithmetic) value of that term and the common difference between reciprocals, so you can check the working or continue the calculation by hand. Results update instantly as you change any of the three inputs. Use it to check homework, verify a pattern you have spotted, or work through problems in algebra, physics or engineering that rely on harmonic progressions, keeping in mind that no term in the sequence can equal zero.

First term

Second term

Position n

Term 5 = 0.2

Reciprocal (arithmetic term)5

Common difference of reciprocals1

The formula

In a harmonic progression the reciprocals are arithmetic. With first term a1 and second term a2, the reciprocal common difference is d = 1/a2 minus 1/a1, the nth reciprocal is 1/a1 + (n minus 1) d, and the nth term is its reciprocal.

Worked example

For first term 1 and second term 0.5, the reciprocals are 1, 2, 3, 4, 5 (common difference 1), so term 5 is 1/5 = 0.2. Enter 1, 0.5, 5 to confirm.

Frequently asked questions

What is a harmonic sequence?

A sequence whose reciprocals form an arithmetic sequence, such as 1, 1/2, 1/3, 1/4.

How do I find a term?

Work with the reciprocals, which increase by a fixed amount, find the one you want, then take its reciprocal.

Is there a simple sum formula?

No neat closed form exists for the sum of a harmonic series; it has to be added up term by term.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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