The Fibonacci sequence is defined by the recurrence relation:

Fn = Fn−1 + Fn−2, where F1 = 1 and F2 = 1. Hence the first 12 terms will be:

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

The 12th term, F12, is the first term to contain three digits.

What is the index of the first term in the Fibonacci sequence to contain 1000 digits?

As discussed in prob. 2, the closed form $i^{\text{th}}$ Fibonacci number is:

$$F_{i} = \frac{(\frac{1+\sqrt{5}}{2})^{i} - (\frac{1-\sqrt{5}}{2})^{i}}{\sqrt{5}}$$

The number of digits is given by: $$\lceil log_{10} F_{i} \rceil = \lceil log_{10} \frac{(\frac{1+\sqrt{5}}{2})^{i} - (\frac{1-\sqrt{5}}{2})^{i}}{\sqrt{5}}\rceil$$

and since $(\frac{1-\sqrt{5}}{2})^{i} \rightarrow 0$ as $i \rightarrow \infty$, we have: $$\lceil log_{10} F_{i} \rceil \approx \lceil i log_{10} \frac{1-\sqrt{5}}{2} - \frac{1}{2} log_{10}(5) \rceil$$


In [1]:
import math

(999 + 0.5*math.log(5)/math.log(10)) / (math.log(((1+5**.5)/2))/math.log(10))


Out[1]:
4781.859270753069

In [2]:
math.ceil(4782 * math.log(((1+5**.5)/2))/math.log(10) - 0.5*math.log(5)/math.log(10))


Out[2]:
1000.0