Question

The ratio Fib(n+1)/Fib(n) as n gets larger is said to approach the Golden Ratio, which is approximately equal to 1.618. Q1: What happens to the inverse of this ratio, Fib(n)/Fib(n+1)? Q2: What number does this quantity approach? Q3: How does this compare to the original ratio?

Answer 1

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Answer 2

Given:

The approximately equal to 1.618.

To find:

Find the inverse of this ratio, Fib(n)/Fib(n+1).and compare to the original ratio.

Step-by-step explanation:

It is approximately equal to $\frac{1}{1.68}$ .This is, in fact 0.618.

The limiting ratio is actually $\frac{\sqrt{5} +1}{2} .$

Try to show that the reciprocal of this is $\frac{\sqrt{5} -1}{2} .$

Then see if you can out these limits

$\frac{Fib(n+1)-Fib(n-1)}{Fib(n)}$

and,

$\frac{Fib(n+1)+Fib(n-1)}{Fib(n)}$