Project Euler Problem 25 Solution
Feb. 25, 2017

What is the index of the first term in the Fibonacci sequence to contain $1000$ digits?
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The question actually asks to find the first fibonacci term, whose number of digits crosses $1000$.

An obvious way to solve it is to brute force all the way to find $F_n$ ($n^{th}$ Fibonacci number) such that $\log_{10}(F_{n})$ is $1000$.
The solution becomes too easy if we use Golden ratio $\phi = 1.61803398875$

After some research on Golden ratio and Fibonacci numbers, we can find out an important fact that

$$\phi^{n-2} \le F_n$$

Now, if a number N has n digits in it then, $\log_{10}N = n-1$. For example, $\log_{10}100 = 2$ and $\log_{10}999 = 2.999$.
So, for $1000$ digits, its logarithm is $999$ because both 100 and 999 have 3 digits.

Using this fact in the above inequality(greater than sign is omitted as we are concerned with the first number to have 1000 digits),

$$\log(\phi^{n-2}) = 999$$ $$or, (n-2) * \log(\phi) = 999$$ $$or, (n-2) * 0.2089 = 999$$

which gives n= 4782, our answer.
Cheers!!