Fibonacci Numbers
The sequence from F₀ to Fₙ, the exact value of Fₙ and the sum of the first terms, with the solution.
What to find:
For the sequence n up to 300, for the number Fₙ n up to 5000. Numbering starts from F₀ = 0.
The Fibonacci calculator lists the sequence from F₀ to Fₙ and finds the number Fₙ and the sum of the first terms. Numbers are exact, with no rounding, even when they have hundreds of digits.
Definition
F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂
The sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Sum
F₀ + F₁ + … + Fₙ = Fₙ₊₂ − 1
Example. 0 + 1 + 1 + 2 + 3 + 5 = 12 = F₇ − 1 = 13 − 1.
The golden ratio
The ratio of neighbouring numbers approaches φ ≈ 1.618, and Fₙ itself is close to φⁿ / √5.
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