Harmonic Sequence Calculator

This harmonic sequence calculator finds the nth term of a harmonic progression, a sequence whose reciprocals form an arithmetic progression. Enter the first two terms and the position you want, and the calculator works out the term. The trick is to flip the sequence over: take the reciprocals, which step up by a constant common difference, find the nth reciprocal, then flip back. Harmonic sequences appear in music, in optics and lens formulas, and in averages of rates. The method, a worked example and the assumptions are below.

First term

Second term

Position n

Reciprocal (arithmetic term)

Common difference of reciprocals

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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