Answer first
What this calculator tells you
Find the nth Fibonacci number, the ratio to the term before it and the sum of the first n terms. See how quickly the sequence grows and how its ratio settles near the golden ratio. Formula: F(n) = F(n−1) + F(n−2), with F(1) = F(2) = 1; the sum of the first n terms = F(n + 2) − 1. At the worked-example inputs, the fibonacci number f(n) is 55. Holding every other input steady, moving position n from 8 to 12 moves the result from 21 to 144.
Transparent method
The formula
See how quickly the sequence grows and how its ratio settles near the golden ratio.
Worked example
Example inputs
How to interpret the result
Each Fibonacci number is the sum of the two before it, so 1, 1, 2, 3, 5, 8 and on to 55 at the tenth place. The sequence grows by about 1.6 times at each step, and the ratio of neighbors closes in on the golden ratio. The sum of the first ten terms is 143, which is always one less than the term two places further on. Shells, petals and stock-chart folklore all cite it.
At the worked-example inputs the fibonacci number f(n) is 55. It rises with position n.
These are exact arithmetic operations; results are limited only by floating-point precision.
Before you rely on it
What to check
Note that the count starts with F(1) = 1 and F(2) = 1 here. Some texts start with F(0) = 0, which shifts every position by one.
The common error
Where people go wrong with fibonacci calculator
Expecting the ratio to hit the golden ratio exactly. It alternates above and below it and only gets closer as the terms grow.
Sensitivity evidence
How position n changes the fibonacci number f(n)
Holding every other input at the worked-example value, moving position n from 8 to 12 moves the fibonacci number f(n) from 21 to 144: a spread of 123, or 224% of the worked-example result.
| Position n | Fibonacci number F(n) | Ratio F(n) ÷ F(n−1) | Sum of the first n terms |
|---|---|---|---|
| 8 | 21 | 1.6 | 54 |
| 9 | 34 | 1.6 | 88 |
| 10worked example | 55 | 1.6 | 143 |
| 11 | 89 | 1.6 | 232 |
| 12 | 144 | 1.6 | 376 |
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Position n | 10 | A whole number from 1 to 76. Beyond that the value is too large to hold exactly. |
| Fibonacci number F(n) | 55 | |
| Ratio F(n) ÷ F(n−1) | 1.6 | |
| Sum of the first n terms | 143 | |
Inputs, definitions and assumptions
Position n
A whole number from 1 to 76. Beyond that the value is too large to hold exactly. The prefilled worked-example value is 10.
How to use this calculator
- 1Verify the inputs. Gather position n from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the fibonacci number f(n) at 55. Store your own version of it as Scenario A.
- 3Test one change. Start with position n, the input with the biggest effect here: moving position n from 9 to 11 takes the fibonacci number f(n) from 34 to 89, a swing of 100% of the worked-example figure.
- 4Check the extremes. At half the example position n (5) the fibonacci number f(n) is 5; at double (20) it is 6,765.
People also ask
Frequently asked questions
How do you calculate fibonacci?
F(n) = F(n−1) + F(n−2), with F(1) = F(2) = 1; the sum of the first n terms = F(n + 2) − 1. At the worked-example inputs the fibonacci number f(n) is 55.
What does the fibonacci result mean?
See how quickly the sequence grows and how its ratio settles near the golden ratio. At the worked-example inputs the fibonacci number f(n) is 55. It rises with position n.
How much does position n change the fibonacci number f(n)?
Holding every other input at the worked-example value, moving position n from 8 to 12 moves the fibonacci number f(n) from 21 to 144, a spread of 123.
What are the limits of this fibonacci calculator?
These are exact arithmetic operations; results are limited only by floating-point precision. The tables on this page test position n only from 8 to 12; a value outside that range is not tabulated here.
If I double position n in the fibonacci calculator, does the fibonacci number f(n) double?
Doubling it from 10 to 20 takes the fibonacci number f(n) from 55 to 6,765, which is 123.00 times the worked-example figure. So the result grows faster than the input does. Halving it to 5 gives 5.
Which inputs change the ratio f(n) ÷ f(n−1) in the fibonacci calculator?
At the worked-example inputs it is 1.6. Position n takes it from 1.6 to 1.6.
Which inputs change the sum of the first n terms in the fibonacci calculator?
At the worked-example inputs it is 143. Position n takes it from 88 to 232.
Why does a computer store everything in binary?
A transistor has two reliable states, on and off, which map directly onto the two binary digits 0 and 1. Decimal would require ten distinguishable voltage levels per digit instead of two, far less reliable to build and read at the speeds modern hardware runs at.
Why do programmers use hexadecimal instead of binary or decimal?
Each hex digit represents exactly four binary digits, so a 32-bit value becomes 8 readable hex characters instead of 32 ones and zeros, with none of decimal's awkward, non-power-of-two grouping. That clean mapping is why memory addresses and color codes are conventionally written in hex.
How do I know what base a number I copied from somewhere is in?
Check for a prefix or explicit label: programming languages commonly use 0x for hexadecimal and 0b for binary, and technical documentation usually states the base outright. The digits alone rarely disambiguate, since 100 is a valid value in every common base.
Sources and evidence
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