Exercises 1. Investigate the convergents for the continued fraction expansion of

Question

Exercises 1. Investigate the convergents for the continued fraction expansion of the golden ra- tio (1 + v ). what do these convergents have to do with the Fibonacci series? Coupled oscillators have a tendency to seek frequency ratios which can be ex- pressed as rational numbers with small numerators and denominators. For exam- ple, Mercury rotates on its axis exactly three times for every two rotations around the sun, so that one Mercurial day lasts two Mercurial years. In a similar way, the orbital times of Jupiter and the minor planet Pallas around the sun are locked in a ratio of 18 to 7 (Gauss calculated in 1812 that this would be true, and observation has confirmed it). This is also why the moon rotates once around its axis for each rotation around the earth, so that it always shows us the same face. Among small frequency ratios for coupled oscillators, the golden ratio is the least likely to lock in to a nearby rational number. Why?

Explanation / Answer

Part - 1

the limit of the ratios of terms between the fibonacci numbers is the golden ratio

1 1 2 3 5 8 13 21..

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.66
8/5 = 1.6
13/8 = 1.625
21/13 ~ 1.61538462...

and so on. One can prove rigorously that this limit approaches (1+sqrt(5))/2 ~ 1.61803399

F[n] is the nth term of the Fibonacci sequence

As n goes to infinity F[n + 1] / F[n] tends towards the golden ratio.

Part - 2

Hey , https://www.math.auckland.ac.nz/~king/745/mediants.pdf

go through this you may not get complete idea but roughly u can understand.

read this, its interesting even though not clear :) hope it helps . you can write an absract answer.

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