Fibonacci Number Calculator: nth Fibonacci and Lucas Numbers, Sum and the Golden Ratio
Fibonacci numbers and the golden ratio.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Enter n.
The calculator adds up the sequence exactly.
It gives the nth term, the sum, the Lucas number and the ratio.
Formulas used
- F_n = F_{n-1} + F_{n-2}
- \sum_{i=0}^{n}F_i = F_{n+2} - 1
- F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5}
Worked example
With the default numbers in the calculator above:
- Fibonacci number F(n): 6765
- Sum of the sequence F(0) to F(n): 17710
- Lucas number L(n): 15127
- Ratio of F(n) to F(n-1): 1.6180339632
- Golden ratio for comparison: 1.6180339887
- Binet formula value: 6765
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What is the Fibonacci sequence?
0, 1, 1, 2, 3, 5, 8, 13 and so on, where every term is the sum of the two before it.
How is it related to the golden ratio?
The ratio of one term to the one before approaches the golden ratio, about 1.618.
What is the Lucas sequence?
It follows the same rule but starts with 2 and 1.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any discrete mathematics text (Fibonacci numbers).
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