EMAT 6680 :: Clay Kitchings :: Assignment 12 :: Fibonacci Numbers

Exploring Fibonacci Numbers

Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1, and f(n) = f(n-1) + f(n-2).

* = values begin with n=2

 n ratios f(n-1) + f(n-2) * f(n)/f(n-1) 0 1 undefined 1 1 1 2 2 2 3 3 1.5 4 5 1.666666667 5 8 1.6 6 13 1.625 7 21 1.615384615 8 34 1.619047619 9 55 1.617647059 10 89 1.618181818 11 144 1.617977528 12 233 1.618055556 13 377 1.618025751 14 610 1.618037135 15 987 1.618032787 16 1597 1.618034448 17 2584 1.618033813 18 4181 1.618034056 19 6765 1.618033963 20 10946 1.618033999 21 17711 1.618033985 22 28657 1.61803399 23 46368 1.618033988 24 75025 1.618033989 25 121393 1.618033989 26 196418 1.618033989 27 317811 1.618033989 28 514229 1.618033989 29 832040 1.618033989 30 1346269 1.618033989

It appears that this sequence converges to approximately 1.61803989...

LetÕs see if we can find an exact value for this approximation.  First, letÕs assume that this sequence does indeed converge to some .  Take a ratio for a high value of n as n approaches infinity.

The sequence might appear as follows:

1, 1, 2, 3, 5, 8, É, a, b, a+b

Consider the ratio:

If the sequence converges, it must converge to some value =b/a (b/a is practically equal to (a+b)/b when n approaches + infinity).  This implies:

Now, use the quadratic formula to solve:

The Fibonacci Sequence converges to .

If we take different ratios, we can conjecture that other such ratios converge as well (to different ).

 n ratios every 2nd ratio every 3rd ratio *f(n-1) + f(n-2) * f(n)/f(n-1) 0 1 undefined 1 1 1 2 2 2 2 3 3 1.5 3 3 4 5 1.666666667 2.5 5 5 8 1.6 2.666666667 4 6 13 1.625 2.6 4.333333333 7 21 1.615384615 2.625 4.2 8 34 1.619047619 2.615384615 4.25 9 55 1.617647059 2.619047619 4.230769231 10 89 1.618181818 2.617647059 4.238095238 11 144 1.617977528 2.618181818 4.235294118 12 233 1.618055556 2.617977528 4.236363636 13 377 1.618025751 2.618055556 4.235955056 14 610 1.618037135 2.618025751 4.236111111 15 987 1.618032787 2.618037135 4.236051502 16 1597 1.618034448 2.618032787 4.236074271 17 2584 1.618033813 2.618034448 4.236065574 18 4181 1.618034056 2.618033813 4.236068896 19 6765 1.618033963 2.618034056 4.236067627 20 10946 1.618033999 2.618033963 4.236068111 21 17711 1.618033985 2.618033999 4.236067926 22 28657 1.61803399 2.618033985 4.236067997 23 46368 1.618033988 2.61803399 4.23606797 24 75025 1.618033989 2.618033988 4.23606798 25 121393 1.618033989 2.618033989 4.236067976 26 196418 1.618033989 2.618033989 4.236067978 27 317811 1.618033989 2.618033989 4.236067977 28 514229 1.618033989 2.618033989 4.236067978 29 832040 1.618033989 2.618033989 4.236067977 30 1346269 1.618033989 2.618033989 4.236067978