- dansmath > fibonacci page
- 1 1 2 3 5 8 13 21 34 55 89 the fibonacci page 2/06
- (c) 2004-06 by Dan Bach (and the entire www.dansmath.com empire!)
- .
- Definition: The Fibonacci numbers start with 1 and 1 . The next Fib #
- is the sum of the previous two : 1 +1 = 2, then 1+2 = 3, 2 + 3 = 5 , etc.
- The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21 , . . . is the Fibonacci Sequence.
- .
- Definition: The Fibonacci numbers start with 1 and 1 . The next Fib #
- We keep track of the position n of the nth Fib # , called Fn :
- .
- F0 = 0 , F1 = 1 , F2 = 1 , F3 = 2 , F4 = 3 , F5 = 5 F6 = 8 , . . . , F10 = 55 , . . .
Some facts about the Fn :
(1) The limit of the ratios Fn+1/Fn is called the Golden Ratio "Phi" - approx. 1.618
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(2) The five-pointed star spinning above is full of golden ratios;
(for example (the distance across) / (the distance between adjacent outer points)
(3) The Golden Rectangle is one where the proportion : 1 / x = (x - 1) / 1
This means that to 3 places , 1 / 1.618 = 0.618 , a surprising decimal indeed!
(4) The Golden Ratio is exactly Phi = (1 + \/ 5) / 2 (and yes this rounds to 1.618)
That's because cross-multiplying the proportion gives x(x - 1) = 1 or
x^2 - x - 1 = 0 ; the quadratic formula gives (1 +\ - sqrt[1 + 4]) / 2 ; use the +
. . . and formulas about the Fn :
(a) The only perfect square after 1 is F12 = 144.
Trust me, it's been checked for huge n's. Is it proved for sure?
(b) The Fibs grow at an approx geometric rate, with ratio Phi of course
F10 = 55 , F11 ~ 55 * 1.618 ~ 88.99 ; F11 = 89 ;
F18 ~ F10 * 1.618^8 ~ 55 * 46.971 ~ 2583.39 . . . is it 2583 or 2584? Need more accuracy on Phi..
(c) The square of any Fn is always 1 away from the product of the two Fibs around it
1^2 = 1 * 2 - 1 ; 2^2 = 1 * 3 + 1 ; 3^2 = 2 * 5 - 1 ; 5^2 = 3 * 8 + 1 ; 8^2 = 5 * 13 - 1
(d) The sum of all the Fibs up to Fn is 1 less than the later Fib Fn+2
1 = 2 - 1 ; 1 + 1 = 3 - 1 ; 1 + 1 + 2 = 5 - 1 ; 1 + 1 + 2 + 3 = 8 - 1 ; 1 + 1 + 2 + 3 + 5 = 13 - 1
(e) The Fib Fnk is always a multiple of Fn ; in other words if n | m then Fn | Fm
F4 = 3 , F8 = 21 , F12 = 144 , etc . . . . F5 = 5 , F10 = 55 , F15 = 610 , etc
(f) The Fib F2n is always Fn times (Fn-1 + Fn+1)
F4 = F2(F1 + F3) ; 3 = 1(1 + 2) . . . . F6 = F3(F2 + F4) ; 8 = 2(1 + 3)
F18 = ? = F9 (F8 + F10) = 34 (21 + 55) = 34 * 76 = 2584.
