Calculator inputs
Example data table
| Index | Reciprocal AP Value | Harmonic Term |
|---|---|---|
| 1 | 2 | 1/2 = 0.5 |
| 2 | 3 | 1/3 ≈ 0.333333 |
| 3 | 4 | 1/4 = 0.25 |
| 4 | 5 | 1/5 = 0.2 |
| 5 | 6 | 1/6 ≈ 0.166667 |
Formula used
A harmonic sequence is a list of non-zero numbers whose reciprocals form an arithmetic progression. If the reciprocal progression starts at a and changes by d, then the harmonic term at position n is:
hₙ = 1 / (a + (n − 1)d)
When only the first two harmonic terms are known, the reciprocal start becomes a = 1/h₁ and the reciprocal difference becomes d = 1/h₂ − 1/h₁. The calculator then generates the full sequence, the requested target term, and the numerical sum of the displayed terms.
For validation, the calculator converts each pasted term into its reciprocal and checks whether consecutive reciprocal differences stay constant within the chosen rounding tolerance.
How to use this calculator
- Select a mode based on the information you already have.
- Enter reciprocal inputs, the first two terms, or a pasted list.
- Choose the start index, number of terms, target index, and precision.
- Press the calculate button to show results below the header.
- Review the summary cards, generated table, and notes.
- Download the current output as CSV or PDF if needed.
Frequently asked questions
1. What makes a sequence harmonic?
A sequence is harmonic when every non-zero term has a reciprocal, and those reciprocals form an arithmetic progression with a constant difference.
2. Can a harmonic term be zero?
No. Zero has no reciprocal, so it cannot belong to a harmonic sequence. The calculator flags any step that would create a zero reciprocal denominator.
3. Why does the calculator use reciprocal values?
That is the defining rule. Harmonic sequences are easiest to generate, analyze, and validate by working first with the arithmetic progression formed by reciprocal terms.
4. What does the target index output show?
It reports the requested term hₙ using the formula based on your selected mode. This helps you study one specific position without manually expanding the whole sequence.
5. Is the generated sum exact?
The displayed sum is numerical for the terms currently generated on screen. It is useful for analysis, but rounding depends on your chosen precision setting.
6. What is validation tolerance?
Validation tolerance is the small allowed gap between reciprocal differences after rounding. It prevents harmless decimal trimming from falsely failing an otherwise harmonic list.
7. Can the sequence increase and decrease?
Yes. The direction depends on the reciprocal progression and on whether denominators approach or move away from zero. Harmonic terms can rise, fall, or change sign.
8. When should I use two-term mode?
Use it when you know the first two harmonic terms but not the reciprocal progression directly. The calculator reconstructs the reciprocal start and common difference for you.