Number Contributions

 We can derive similar results for sums of Fibonacci decimal type sequences where we combine multiples of terms. The first case again is when we combine multiples of the last term (a) with the first term. Again the simplest example is when a = 2 so that we repeatedly combine twice the last term with the previous term in deriving our sequence. This results in the series 1, 2, 5, 12, 29, 70, 169, 408, …. The sum of the corresponding decimal point series = the reciprocal of 79. Thus  1/79 = .01 + .002, + .0005 + .00012 + .000029 + .0000070 + .00000169 + .000000408 + …. Thus whereas the Fibonacci point series = 100 - 10 - 1 (where a = 1), the Pell is 100 - 20 - 1 (where a = 2). Thus in general terms where we use a multiple of the second term (i.e. a>1 ) , the sum of the point series is given by  100 - 10a - 1 . Thus where a = 3, the sum of the point series = 100 - 30 - 1 = 69. This series where we continually combine 3 times the last with the second last term is 1, 3, 10, ...

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 ...

 We will now consider sequences where a multiple of the last term is combined with the second last term to be alternately reversed with the same multiple of the second last term to be combined with the last. The simplest case is where (starting with 0, 1) we combine twice the last term with the second last term and then alternately combine twice the second last with the last term. This leads to the following sequence 0, 1, 2, 4, 10, 18, 46, 82, 210, 374, 958, 1706, 4370 ,…. Again this leads to two ratios emerging The larger is approximated best here by  4370/1706 = 2.5615… The smaller is approximated by  1706/958 = 1.7807.. There is a simple relationship as between these two ratios in that the larger (2.5615) =  2*1.7807. - 1. Now the square of 2,5615... = 6.5615… Therefore k(squared) = k + 4 . This enables us to solve the value of k from the equation  k 2  - k - 4 = 0  = (square root of 17 + 1)/2 This in turn enables us to solve the smaller ratio as a...

 In the previous article, the relevant equation for multiple combinations (a and b) of the last and second last terms in the Fibonacci type sequence was  x 2  - ax - b = 0  and thereby unitary in the x 2  term. However we can generalize this first term also by considering another fascinating situation where we alternately vary the way in which we combine terms. Thus is we alternately combine twice the last term with the previous term  (a= 2, b = 1)  , and then the last term with the previous  (a = 1, b = 1) , we generate the following sequence 0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, …. We now obtain two ratios when we divide a term by the previous term. Thus the ratio of the last given terms in this series is  780/571 = 1.36602… However starting with the previous term, the ratio is  571/209 = 2.73205… Now the latter value approximates closely to the value of  (the square root of 3) + 1 , whereas the former value approxima...

    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,...

The procedure adopted in the article can be summed up through a well-ordered series of fascinating connections between phi and the respective terms of the Lucas series 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 621, 943, …. Thus Phi - 1/Phi = 1 (the first term of Lucas Series) Phi 2  + 1/Phi 2  = 3 (the second term of Lucas series) Phi 3  - 1/Phi 3  = 4 (the third term of Lucas Series) Phi 4  + 1/Phi 4  = 7 (the fourth term of Lucas Series) Phi 5  - 1/Phi 5  = 11 (the fifth term of Lucas Series) So in general format Phi n  (+ or -) 1/Phi n  = T n  (where Tn represents the nth term of the Lucas Series). When n is even we add the respective values of Phi. When n is odd, we subtract the respective values of Phi. Thus for example T 11  = Phi 11  - 1/Phi 11  = 199 (which is the 11 th  term in the series).

  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