## anonymous one year ago Two large numbers of the Fibonacci sequence are F50 = 12,586,269,025 and F51 = 20,365,011,074. What is the approximate value of the quotient ? f51/50

1. anonymous

A. 1.618 B. 0.618 C. 2.618 D. 2 E. 3.142

2. anonymous

@Mehek14

3. anonymous

we have that $$F_n\approx\frac1{\sqrt5}\varphi^n$$ for large $$n$$ where $$\varphi=\frac12(1+\sqrt5)\approx1.618$$

4. anonymous

so $$F_{n+1}/F_n\approx \varphi\approx 1.618$$ for large $$n$$; in this case, $$n=50$$ is sufficiently big for this approximation to be 'good'

5. anonymous

so it =1.618

6. anonymous

????? @oldrin.bataku

7. anonymous

check for yourself if you have a calculator; even just 3 digits of each are enough to give a decent approximation

8. anonymous

$$F_{51}/F_{50}\approx 204/126=1.619\approx1.618$$