G-Fact 18 | Finding nth Fibonacci Number using Golden Ratio

We have discussed different methods to find nth Fibonacci Number.

Following is another mathematically correct way to find the same.

nth Fibonacci Number :
 F(n) = \left \lfloor \frac{\varphi^n}{\sqrt5} + \frac{1}{2} \right \rfloor, n >= 0
Here φ is golden ratio with value as (\sqrt 5+1)/2
The above formula seems to be good for finding nth Fibonacci Number in O(Logn) time as integer power of a number can be calculated in O(Logn) time. But this solution doesn’t work practically because φ is stored as a floating point number and when we calculate powers of φ, important bits may be lost in the process and we may get incorrect answer.


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