0
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. Large results can also exceed ordinary exact numeric storage, so use modest indices for worked examples that require an exact integer.
| 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 a simple algorithm and studying how a rule plus starting values defines a sequence. The calculator lets you compare an independently calculated term with an iterative result.
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 repeatedly adds the last two values using the server’s ordinary numeric representation. That is efficient for small examples, but it is not an arbitrary-precision sequence generator. At sufficiently large indices, exact integer digits can be lost or the result can become too large to represent. Formatting the result with commas improves readability without extending that numeric precision.
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.
Mini Calculator provides online calculators for multiple niches including mathematical, financial, Health, informative, Chemistry, physics, statistics, and conversions. You can use any calculator for free without any limits.