Fibonacci Calculator
Find the nth Fibonacci number and the full sequence up to that term, plus the running sum.
How it works
F(1) = 1, F(2) = 1, F(n) = F(n−1) + F(n−2) · sum of first n terms = F(n+2) − 1
The Fibonacci sequence starts 1, 1 and builds every later term by adding the two before it. It appears in phyllotaxis, population models and computer science, and the ratio of consecutive terms creeps ever closer to the golden ratio φ ≈ 1.618. This calculator uses exact big-integer arithmetic so even large terms are computed without rounding error.
Worked example
Counting 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, the 10th Fibonacci number is 55. The first ten terms add up to 143 — which matches the shortcut F(12) − 1 = 144 − 1 = 143.
Sources: Fibonacci sequence — Wikipedia
Frequently asked questions
How is the sequence indexed here?
This calculator is 1-indexed with F(1) = 1 and F(2) = 1, so term 10 is 55. Some references start with a 0 (F(0) = 0), which shifts every position by one — check the convention when comparing.
What is the sum of the first n Fibonacci numbers?
It equals F(n+2) − 1. The first ten terms sum to F(12) − 1 = 143, a handy identity that avoids adding them one by one.
How do Fibonacci numbers relate to the golden ratio?
Divide any term by the one before it and the quotient homes in on φ ≈ 1.618. For example 55 ÷ 34 ≈ 1.6176, already very close.
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This calculator is for educational purposes. Double-check important results before acting on them.