Fwd: . What if the ratio of consecutive Fibonacci Numbers did converge . 07 apr 2017 fri 0931 am PDT .

[h.s.n...@gmail.com]

On Friday, April 7, 2017 at 10:28:01 AM UTC-7, h.s.n...@gmail.com wrote:

. What if the ratio of consecutive Fibonacci Numbers did converge . 07 apr 2017 fri 0931 am PDT .
. date 07 apr 2017 fri 0931 am PDT .
. topic . What if the ratio of consecutive Fibonacci Numbers did converge .
. Fibonacci numbers are defined by the sequence … a b a+b … starting with 1 1 . Consecutive Fibonacci numbers are obtained by adding the last two numbers to get the next number . The first few of them may be enumerated as 1 1 2 3 5 9 13 21 33 54 … I read a post in this internet newsgroup soc.culture.indian dated 03 apr 2017 by DrJM titled . Using Fibonacci Numbers to convert from miles to kilometers and vice versa . and discussed at the website

http://www.catonmat.net/blog/using-fibonacci-numbers-to-convert-from-miles-to-kilometers/

. The discussion is about the ratio of consecutive Fibonacci numbers converging to a number 1.6 that is also the ratio relating 1 mile to 1 kilometer . Upon further investigation I find that it is possible to mathematically arrive at the exact ratio . If the forward ratio is defined as Fn+1 divided by Fn and the reverse ratio is defined as Fn divided by Fn+1 where Fn and Fn+1 are consecutive Fibonacci numbers then we can see that for the sequence a b a+b the forward ratio would be  ( a + b ) / b or 1 + ( a / b ) or 1 + x where x is the fraction a / b . The reverse ratio for these numbers would be a / b or x . Now we may state that if the two ratios converge then as the numbers tend to infinity the reciprocal of the forward ratio would approach the reverse ratio . This means 1 / ( 1 + x ) = x for large numbers in the sequence .This is a quadratic equation ( x squared ) + x - 1 = 0  with roots x = ( -1 ( + or - ) sqrt ( 5 ) ) / 2 . That gives x = 0.61803398875 since the negative root is not meaningful for the sequence . You may verify that ( 1 / x ) = 1.61803398875 = 1 + x as I found using the google.com online calculator . That means if the ratio of consecutive Fibonacci numbers converges then it would converge to a forward ratio of 1.61803398875 and a reverse ratio of 0.61803398875 . end of message . Please send me your comments if any . My email is h.s.n...@gmail.com . Thank you . Hemarajan Shankaranarayanan Nair .