What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding terms. Starting from 1 and 1, the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. It was described by the Italian mathematician Fibonacci in the 13th century and appears in nature in the spiral arrangements of seeds, petals, and shells.

What is the formula for the nth Fibonacci number?

The direct (closed-form) formula is known as Binet's formula: F(n) = (phi to the n minus psi to the n) divided by the square root of 5, where phi = (1 plus square root of 5) divided by 2 (approximately 1.618) and psi = (1 minus square root of 5) divided by 2 (approximately minus 0.618). Because psi has absolute value less than 1, its contribution vanishes for large n and F(n) is very close to phi to the n divided by the square root of 5.

What is the connection between Fibonacci numbers and the golden ratio?

The ratio of consecutive Fibonacci numbers, such as 8 divided by 5 or 55 divided by 34, converges to the golden ratio phi as the terms get larger. By the time you reach F(20) divided by F(19) the ratio is accurate to more than six decimal places. This is why the Fibonacci sequence and the golden ratio are so closely linked in both mathematics and design.

Fibonacci Calculator

The Fibonacci sequence is one of the most famous number patterns in mathematics. Each term is formed by adding the two terms that come before it: the sequence starts 1, 1, 2, 3, 5, 8, 13, 21, and keeps going indefinitely. The nth Fibonacci number, written F(n), is your position in this sequence. So F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3, F(5) = 5, and so on. What makes the sequence remarkable is how it appears throughout nature, from the spiral arrangements of seeds in a sunflower to the branching of trees and the growth of shells, and how it is intimately linked to the golden ratio phi. As you go further along the sequence, the ratio of any term to the one before it gets closer and closer to phi, approximately 1.61803. By the 20th term the approximation is accurate to six decimal places, and by the 30th term it is accurate to more than twelve. You enter the position n, from 1 up to 70, and the calculator returns F(n), the complete sequence from F(1) to F(n), the ratio of the last two terms as an approximation of phi, and the number of digits in F(n). The sequence is computed iteratively to avoid floating-point rounding issues for small to medium values of n. This tool is useful for students studying sequences and series, for anyone curious about the golden ratio, and for programmers testing Fibonacci implementations. Values above n=70 can exceed the safe integer range for standard JavaScript arithmetic.

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Position n

F(10): the 10th Fibonacci number

F(n)/F(n-1) ratio1.617647

Digits in F(n)2

Sum of sequence143

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Sequence is computed iteratively. Reliable for n up to 70. Above that, JavaScript number precision may cause rounding.

How it works

The calculator builds the Fibonacci sequence iteratively: F(1) = 1, F(2) = 1, and for each subsequent term F(k) = F(k-1) + F(k-2). This avoids the rounding errors that can accumulate with Binet's closed-form formula at large n. The ratio F(n)/F(n-1) is the best rational approximation to the golden ratio phi achievable with consecutive Fibonacci numbers. The sum of the first n Fibonacci numbers equals F(n+2) minus 1, a useful identity that the calculator verifies by summing all displayed terms. Digits in F(n) are counted by converting the number to a string and measuring its length.

Worked example

Using the default n = 10: the sequence is built step by step as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Therefore F(10) = 55. The ratio of the last two terms is 55 divided by 34 = 1.617647, already close to phi = 1.61803. The sum of all ten terms is 1+1+2+3+5+8+13+21+34+55 = 143, which also equals F(12) minus 1 = 144 minus 1 = 143.

Related calculators

Golden Ratio Calculator: split any length using phi, the limit of consecutive Fibonacci ratios.
Arithmetic Sequence Calculator: nth term and sum of an arithmetic progression.
Geometric Sequence Calculator: nth term and sum of a geometric progression.
Summation Calculator: evaluate sigma expressions with custom formulas.

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