Fibonacci Sequence Calculator
This Fibonacci sequence calculator gives any term exactly — every digit, however far along you go — and shows which counting convention it is using instead of quietly picking one.
The sequence itself is a single rule: every term is the sum of the two before it. From 0 and 1 that gives 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Nothing else about it is complicated; the difficulties are all about counting and about size.
Where the Counting Starts
Ask two textbooks for the 10th Fibonacci number and you may get two answers, and neither of them is a mistake. The reason is that F₀ and F₁ are both in use as names for the first term.
Counting from 0, the sequence is F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, …
Counting from 1, it is F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5, … — the same numbers with the zero dropped.
Here is the part worth holding on to: the two agree at every position from 1 onwards. F₁₀ is 55 under both, and so is every other term you are likely to be asked for. What they really differ on is whether an F₀ exists at all.
So where do wrong answers come from? Almost always from a third thing that is neither convention: writing out 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and then counting the zero as the first term, which makes the tenth term 34. It is an easy slip and it is the reason this calculator puts the convention on a switch you can see, and shows the list it is actually using.
How Big These Numbers Get
Fibonacci numbers grow faster than people expect, and they outgrow ordinary computer arithmetic sooner than that.
Whole numbers stop being held exactly past about sixteen digits in the usual way computers store them — and F₇₉, at 14,472,334,024,676,221, is already past that line. From roughly there on, a calculator that works the ordinary way starts returning answers that are a few units out, without any warning that anything has gone wrong.
It gets worse with a shortcut that looks appealing. Binet's formula gives the nth Fibonacci number in closed form using the golden ratio and a square root of 5, and it is often reached for because it needs no loop. In ordinary decimal arithmetic it stops being reliable somewhere in the seventies — and at least one calculator online states outright that it uses Binet up to n = 500, which means quietly wrong digits across most of that range.
This page just adds, in whole numbers with no length limit. F₁₀₀ = 354224848179261915075, all twenty-one digits. F₅₀₀ runs to 105 digits and every one of them is right.
The Golden Ratio, and Why It Turns Up
Divide each term by the one before it and the answers settle down very quickly:
8/5 = 1.6 13/8 = 1.625 21/13 ≈ 1.6154 34/21 ≈ 1.6190 89/55 ≈ 1.61818
They are closing in on φ = 1.6180339887…, the golden ratio. This is not a curiosity bolted on to the sequence; it follows from the rule. If the ratio of consecutive terms settles on some value x, then since each term is the sum of the two before it, x must satisfy x² = x + 1. The positive solution of that quadratic is exactly (1 + √5) / 2.
Which also means the ratios of the Lucas numbers, or of any sequence built the same way, close in on the same value — the starting pair changes the terms but not the rule they obey.
Lucas Numbers, and Any Other Start
Because the rule needs two starting values, changing them gives a different sequence with identical behaviour. The best known is the Lucas sequence, from 2 and 1:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76
Its 10th term is 76. Put any two whole numbers in the first two boxes here — including negatives — and you get that sequence instead, worked out the same way. Leave them empty and you get the ordinary Fibonacci numbers.
One related question the calculator answers directly: is a given number a Fibonacci number at all? There is a clean test — a whole number n qualifies exactly when 5n² + 4 or 5n² − 4 is a perfect square. For 4, neither is, so 4 is not one. For 144, one is, and it sits at position 12.
How to Use This Fibonacci Sequence Calculator
Set the count from switch first, then fill in what you know.
For the ordinary sequence, leave the first two boxes empty and give a position. For Lucas numbers, put 2 and 1 in them. For anything else, put your own two whole numbers there.
To go the other way, leave the position empty and type a value into the last box — you get the position back, or a plain statement that the number does not appear. Since 1 genuinely occupies two positions, that case is answered with both rather than with whichever comes first.
Fill in both a position and a term and they are checked against each other, which is the quickest way to settle an argument about the counting.
| count from | Whether the first term is called F₀ or F₁. The sequence moves with it |
| first term | Leave empty for the ordinary sequence. A whole number |
| second term | The other starting value. Give both or neither |
| position n | Which term you want |
| term | A value to look up, if you would rather find its position |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign, for a starting pair that is negative |
| ⌫ / AC | Delete one character, or clear everything |
| = | Works it out |
The working underneath shows the run itself, the answer with its digit count, and the consecutive ratios closing in on φ — as exact fractions with their decimals beside them, so you can watch it happen rather than take it on trust. If you would like the total of a run of terms rather than one of them, the sigma notation calculator adds any list exactly.
Fibonacci FAQ
What is the Fibonacci sequence?
Does the Fibonacci sequence start at 0 or at 1?
What is the 100th Fibonacci number?
How can you tell if a number is a Fibonacci number?
What are the Lucas numbers?
Why do Fibonacci ratios approach the golden ratio?
Add the last two, watch the ratio settle, and check which end you are counting from. Everything else about this sequence — the golden ratio, the Lucas numbers, the twenty-one-digit hundredth term — comes out of that one rule.
