Number Magic (Series of Contributions on Fibonacci Sequence)

 Number Magic (Series of Contributions on Fibonacci Sequence)


1. Phi in the Sky

2. Recent Addendum

3. Extended Fibonacci -Type Sequence

4. Fibonacci Extensions - Alternating Sequences

5. Fibonacci Extensions - Alternating Sequences

6. Remarkable Binary Connections

7. Further Point Series Connections

Comments

Post a Comment

Popular posts from this blog

Extended Fibonacci-Type Sequences

    34,  55,  89,  144 , 233,  377,  610,  987,…. etc,  the ratio of successive terms  n/(n - 1) approximates to phi . (For example the ratio of 987 and 610 which is 1.61803… approximates phi correct to 5 decimal places). The value of phi which = (1 + square root of 5)/2, is the positive solution for x to the simple algebraic equation  x 2  - x - 1 = 0 . This represents a special case of the more general equation  x 2  - ax - b = 0  (where both a and b = 1). So to generate the Fibonacci sequence we keep adding the last term (a) to the second last term (b) in the sequence to generate the next term. So in the above sequence 987 =  610 + 377 (i.e. a and b = 1). However we can combine different multiples of a and b to generate different series with their own unique features. For example the Pell Series is derived from adding  2a + b . This results in the sequence 0,  1,  2,  5,  12,...
Read more

Remarkable Binary Connections

As has been pointed out elsewhere there is a fascinating link between the reciprocal of 89 and the sum of the decimal point sequence made up Fibonacci terms so that 1/89 = .01 + .001 + .0002 + .00003 + .000005 + .0000008 + .00000013 + .000000021 + .0000000034 + … What is involved here is a well-ordered numerical pattern whereby in base 10,  89 = 100 - 10 - 1. ( 89 is a term in the Fibonacci Sequence ) This can be then generalized for any base Thus in  base 8  for example,  100 - 10 - 1 = 67 . (67 = 55 base 10 is a term in the Fibonacci Sequence). Therefore the reciprocal of 67 is equal to the same decimal point sequence of Fibonacci terms (expressed in base 8). Thus 1/67 = .01 + .001 + .0002 +.00003 + .000005 + +.0000010 + .00000015 + .000000025 + .0000000042 + … This relationship of course also holds in base  2  where  100 - 10 - 1 = 1 . So we have the remarkable result that sum of this decimal point sequence of Fibonacci terms = 1. Converting denary ...
Read more