Fibonacci Calculator

The Fibonacci sequence is an ordered list of integers in which each term after the first two equals the sum of the two terms immediately before it. Using the common modern convention, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, and continues without end. The value at position n is written F(n), with F(0) equal to 0 and F(1) equal to 1. This calculator follows that zero-based convention, so position 10 produces 55. Choose Nth Fibonacci number when you need one exact term. The calculator uses an iterative recurrence rather than a rounded decimal approximation, and JavaScript big integers preserve every digit even for positions far beyond the range of ordinary floating-point numbers. Choose First N numbers when you want to inspect the progression from F(0) onward. Sequence generation is limited to 100 displayed terms so the result remains readable, while the single-position mode supports much larger positions. Fibonacci numbers appear throughout mathematics and computer science. They provide classic examples for recurrence relations, induction, dynamic programming, algorithm analysis, and generating functions. Consecutive positive terms also produce ratios that approach the golden ratio, approximately 1.6180339887. This convergence is not exact at a finite position, but the approximation improves rapidly as the terms increase. The sequence is also related to Pascal's triangle, tiling problems, branching patterns, and many counting arguments. The calculator deliberately starts at zero because that indexing aligns naturally with the recurrence and is widely used in programming. Some school texts instead label 1 as both the first and second term. Those books describe the same numerical pattern with shifted position labels, so always check the convention before comparing an answer. For example, this tool reports F(6) as 8, while a one-based list may call 8 the sixth displayed positive term. To verify a result manually, start with 0 and 1 and repeatedly add the latest two values. For F(7), the additions create 1, 2, 3, 5, 8, and finally 13. This simple rule makes the sequence approachable for students while its deeper properties remain useful in number theory, combinatorics, finance models, and algorithm design.