Find an exact Fibonacci number by index, including values larger than native integer arithmetic can store.
Complete the required fields and select the calculate button.
Indices run from 0 through 5,000. Values are calculated as decimal strings so large terms remain exact.
The Fibonacci sequence starts with 0 and 1, then builds each new term by adding the two preceding terms. Its opening values are 0, 1, 1, 2, 3, 5, 8 and 13. This calculator returns a term at a chosen index, rather than adding every term up to that index.
The index convention matters. On this page, the initial zero is F₀ and the following one is F₁. That means an input of 6 gives F₆ = 8, even though 8 is the seventh entry when you count the displayed list from its first position.
Fractional indices and negative-index extensions are outside this form’s intended use. The calculator uses decimal-string addition so every displayed digit remains exact throughout the supported range.
| Quantity | Explanation |
|---|---|
| n | The sequence index. It is a count, so enter a nonnegative whole number. |
| F₀ and F₁ | The fixed starting values 0 and 1. They define the particular sequence generated here. |
| Fₙ | The value at index n, obtained by repeated addition. The output is not the index itself or a cumulative sum. |
To calculate F₇, start with F₀ = 0 and F₁ = 1. Then F₂ = 0 + 1 = 1, F₃ = 1 + 1 = 2, F₄ = 1 + 2 = 3, F₅ = 2 + 3 = 5, F₆ = 3 + 5 = 8, and F₇ = 5 + 8 = 13.
If a calculation gives F₈ = 21 and F₉ = 34, the next value must be 21 + 34 = 55. This local check catches many copying errors. To establish the whole sequence, however, also check the starting values: other sequences can follow the same addition rule with different seeds.
At index 5, the term is F₅ = 5. The sum from F₀ through F₅ is 0 + 1 + 1 + 2 + 3 + 5 = 12. If a question asks for a total, returning 5 answers the wrong question even though it is a correct Fibonacci term.
This sequence is useful for practicing recurrence relations, testing an algorithm and studying how a rule plus starting values defines a sequence. The previous and next outputs provide a direct local check: adding the previous term to the result must produce the next term.
Some teaching materials begin the displayed list with 1, 1 and call those the first and second terms. Their nth term can therefore differ from an index that starts at zero. Write an explicit label, such as F₆, alongside the answer rather than relying only on “the sixth number.”
The implementation adds nonnegative integer strings one digit at a time rather than relying on a fixed-size server integer. Commas are inserted only after the calculation. This keeps every displayed digit exact through the supported maximum index of 5,000.
The first one is a starting value, F₁ = 1. The next term follows the recurrence: F₂ = F₁ + F₀ = 1 + 0 = 1. Repeated values at different indices are allowed.
Yes. This page uses F₀ = 0. Its presence also makes the recurrence produce F₂ = 1 from the two starting values. When comparing lists, check whether the source includes this zero.
No. Early ratios change: 2/1 = 2, 3/2 = 1.5 and 5/3 is about 1.667. For large positive indices the ratios approach the golden ratio, but finite terms generally do not have exactly that ratio.
Yes. Starting with 2 and 1 gives 2, 1, 3, 4, 7 and 11, which is different from the Fibonacci sequence. Both the recurrence and the starting values are necessary to specify a sequence.
With the indexing used here, F₀ + F₁ + … + Fₙ = Fₙ₊₂ − 1. For n = 5, the sum is 12 and F₇ − 1 = 13 − 1 = 12. This identity is a check on a cumulative sum, not a change to what the calculator returns for an individual index.