Fibonacci Sequence Generator

The sequence starts 0, 1 and every term after is the sum of the two before it. Enter how many terms you want and the tool lists them all, with the final term, the running sum, and the ratio of the last two — the number that converges on the golden ratio φ ≈ 1.618034. Everything is computed in arbitrary-precision integers, so the hundredth or thousandth term is exact rather than rounded.

That exactness is the whole reason to use a tool rather than a spreadsheet. Fibonacci numbers outgrow double-precision arithmetic quickly: F(78) is 8,944,394,323,791,464, which still fits in the 53 bits a double can hold exactly, and F(79) is 14,472,334,024,676,221, which does not. From there on, a calculator working in floating point returns numbers that look right and are wrong in the last digits, with no warning. F(100) is 354,224,848,179,261,915,075, twenty-one digits, and F(1000) has 209.

The convergence to φ is fast and worth watching in the output. The error alternates sign and shrinks by a factor of φ² ≈ 2.618 each step: the ratio is off by about 1.5 × 10⁻⁴ at the tenth term, 10⁻⁸ at the twentieth and 10⁻¹² at the thirtieth. By term thirty it agrees with φ to more digits than a double can represent, which is why the ratio column stops changing long before the sequence does.

Two properties that are more than curiosities. Fibonacci numbers modulo any number are periodic — the last digit repeats with a period of exactly 60, so F(n) and F(n+60) always end in the same digit, which is a quick sanity check on a long list. And because φ is close to the number of kilometres in a mile, consecutive terms convert between them: 5 miles is about 8 km, 8 is about 13, 13 is about 21. The ratios 1.6, 1.625 and 1.615 bracket the true 1.609, so the trick is accurate to within about one percent from the fifth term onward.