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Problem 25

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$1000$-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n-1} + F_{n-2}$, where $F_1 = 1$ and $F_2 = 1$.

Hence the first 12 terms will be:
$$\begin{aligned}
F_1&=1\\
F_2&=1\\
F_3&=2\\
F_4&=3\\
F_5&=5\\
F_6&=8\\
F_7&=13\\
F_8&=21\\
F_9&=34\\
F_{10}&=55\\
F_{11}&=89\\
F_{12}&=144\\
\end{aligned}$$

The $12$th term, $F_{12}$, is the first term to contain three digits.

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

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$1000$位斐波那契数

斐波那契数列是按如下递归定义的数列：
$F_n = F_{n-1} + F_{n-2}$，且$F_1 = 1$，$F_2 = 1$。

因此斐波那契数列的前12项分别是：
$$\begin{aligned}
F_1&=1\\
F_2&=1\\
F_3&=2\\
F_4&=3\\
F_5&=5\\
F_6&=8\\
F_7&=13\\
F_8&=21\\
F_9&=34\\
F_{10}&=55\\
F_{11}&=89\\
F_{12}&=144\\
\end{aligned}$$

在斐波那契数列中，第一个包含三位数字的是第$12$项$F_{12}$。

在斐波那契数列中，第一个包含$1000$位数字的是第几项？

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