The rule
F(n) = F(n−1) + F(n−2), starting from 0 and 1
Why it works
Each Fibonacci number is the sum of the two before it. That single rule produces a sequence that turns up in botany, in art and in the analysis of algorithms, and whose consecutive ratios approach the golden ratio.
How to do it by hand
- Start with 0 and 1
- Add the last two to get the next
- Repeat until you reach the term you want
- Divide any term by the previous one to see the ratio converge
What is worth knowing
The convergence on the golden ratio, about 1.618, is exact in the limit and fast in practice: by the tenth term the ratio is already right to three decimals. Fibonacci introduced the sequence to Europe in 1202 with a problem about breeding rabbits, though Indian mathematicians had described it centuries earlier. The botanical connection is real but often overstated: sunflower seed spirals and pine cone scales do follow Fibonacci counts, because that packing is efficient, but plenty of plants do not. Beyond term 78 the numbers exceed what a double can hold exactly, which is where this calculator stops.