Fibonacci Calculator
Find any term from – 250 to 250 using the Fibonacci recurrence, with optional custom starting values and an exact downloadable workbook.
Sequence inputs
Index is an integer; starting values accept exact decimal notation.
Required integer from – 250 to 250. Example: 10.
Modify F₀ and F₁
Required decimal, up to 12 fractional places. Example: 0.
Required decimal, up to 12 fractional places. Example: 1.
Live result
The recurrence is evaluated exactly for the entered decimals.
Nearby sequence terms
| Index | Exact term |
|---|
How to use this Fibonacci calculator
What this calculator does
This calculator finds the term at an integer index in a Fibonacci-type sequence. The classical sequence starts with F₀ = 0 and F₁ = 1, then follows the recurrence Fₙ = Fₙ₋₁ + Fₙ₋₂. You can keep those familiar starters or replace them with your own decimal values. The tool also works backward for negative indices by rearranging the same recurrence. It is intended for exact sequence arithmetic over the supported index range, not for predicting natural patterns, market prices, or any other external phenomenon. For the standard mathematical definition and common identities, see the Encyclopedia of Mathematics entry on Fibonacci numbers.
When to use it
Use it when you need to check a textbook exercise, verify a program or spreadsheet that generates a recurrence, inspect a negative-index term, or explore how different initial values change a second-order additive sequence. It is also useful when you want an exact workbook record of the inputs and the nearby terms instead of copying rounded values by hand.
How to calculate
- The calculator opens with a complete demonstration: Index n = 10, First term (F₀) = 0, and Second term (F₁) = 1. The first result is F₁₀ = 55, and the Download Excel button is immediately available for this example.
- Replace Index n with the integer position you need. Open Modify F₀ and F₁ if you want nonstandard starters, then enter First term (F₀) and Second term (F₁). Results update live; there is no separate Calculate button.
- Read the n-th term first, then use Previous term, Next term, the ratio, the recurrence check, and the nearby-terms table to inspect context. Choose Download Excel to export the current validated state. Reset clears the demonstration data rather than restoring it; after Reset, export remains disabled until all three required inputs are valid again.
Input guide
Output guide
n-th term (Fₙ) is the exact value at the requested index. Previous term (Fₙ₋₁) and Next term (Fₙ₊₁) are exact neighboring values generated from the same starters. Next/current ratio is a decimal comparison of Fₙ₊₁ to Fₙ when Fₙ is nonzero; for the classical positive sequence it tends toward the golden ratio, a relationship summarized by Wolfram MathWorld's Fibonacci number reference. If the current term is zero, the ratio is shown as not defined rather than forcing a divide-by-zero result. Recurrence check confirms that the exact identity Fₙ₊₁ = Fₙ + Fₙ₋₁ holds. The formula line writes that identity with the current values. The Nearby sequence terms table lists up to eleven exact index/value pairs around n; it is a context table, not an approximation.
Worked example
With the startup values F₀ = 0 and F₁ = 1, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. At n = 10, the previous term is F₉ = 34 and the requested term is F₁₀ = 55. Applying the recurrence once more gives F₁₁ = 55 + 34 = 89, so the displayed Next/current ratio is 89 ÷ 55 = 1.61818182 after display rounding. The workbook uses the same canonical values. The standard sequence is cataloged as OEIS A000045, which is useful when checking known terms independently.
How the calculation works
The defining rule is simple: every term is the sum of the two immediately before it. For positive indices, the calculator starts at F₀ and F₁ and repeatedly adds forward until it reaches the requested neighborhood. For a negative index, the same equation is rearranged to Fₙ₋₂ = Fₙ – Fₙ₋₁, so no separate sign rule or estimated extension is needed. This also works with custom starters, including exact terminating decimals.
Although a closed form involving the golden ratio can compute the classical sequence, floating-point implementations may round large values. This calculator instead represents each entered decimal as an integer numerator over a power-of-ten scale and performs the recurrence with integer arithmetic. That means a terminating decimal starter stays exact throughout the supported range. The ratio output is intentionally a decimal summary; it is not used to calculate the sequence itself.
Interpreting custom starters and negative indices
Any two starting numbers determine a unique additive recurrence. For example, F₀ = 2 and F₁ = 3 produces 2, 3, 5, 8, 13 and so on. Going backward produces F₋₁ = 1 because 3 = 2 + 1. With the classical starters, backward terms alternate in sign according to the well-known negafibonacci identity, but custom starters need not follow that same magnitude pattern. The recurrence itself is the dependable rule in both directions.
The supported range of – 250 to 250 keeps the interface aligned with a practical 250-term calculation window while still allowing exact large integer values. If you are validating software that needs far larger indices, use arbitrary-precision algorithms designed for that scale and be explicit about whether your result must be exact or merely a floating approximation.