Calculus Solver Fibonacci sequence list
The Fibonacci sequence starts 0, 1 and then each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The rule is F(n) = F(n-1) + F(n-2). Dividing each term by the one before it approaches the golden ratio, about 1.618.
The first thirty terms
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229.
That is F(0) through F(29) using the modern convention that starts at 0. Some older books start at 1, 1, which shifts every index by one: it is a labelling difference, not a different sequence.
The rule, and how fast it grows
F(n) = F(n-1) + F(n-2), with F(0) = 0 and F(1) = 1. Every term needs the two before it, which is what makes the sequence recursive.
Growth is exponential: each term is roughly 1.618 times the one before, so the sequence passes a million by F(30) and a billion by F(45). Computing F(50) by naive recursion takes billions of calls, which is why it is the standard example for teaching memoisation.
The golden ratio
Divide consecutive terms and the answers close in on φ = 1.6180339887...: 5/3 = 1.667, 13/8 = 1.625, 89/55 = 1.618. The ratio alternates above and below the limit while the gap shrinks.
That limit is exact in the tail, not approximate: φ is the positive solution of x² = x + 1, which is the same relationship the sequence obeys one step at a time.
The sequence and the ratio between terms
| n | F(n) | F(n)/F(n-1) |
|---|---|---|
| 5 | 5 | 1.667 |
| 8 | 21 | 1.615 |
| 11 | 89 | 1.618 |
| 15 | 610 | 1.618 |
| 20 | 6765 | 1.618 |
Questions
Does the Fibonacci sequence start at 0 or 1?
Both conventions are in use. The modern one starts 0, 1, 1, 2; the older one starts 1, 1, 2, 3. The numbers after the start are identical, only the indexing shifts.
What is the 20th Fibonacci number?
6765, counting from F(0) = 0. If your book starts at F(1) = 1, its 20th term is 6765 as well but its indices differ by one further along.
What is the connection to the golden ratio?
The ratio of consecutive Fibonacci numbers converges to the golden ratio, about 1.618. The further along the sequence you go, the closer the ratio gets.
More calculus reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the rule that justified each line named.
