INTERESTING RELATIONSHIP



 
 

In the 2-term Fibonacci case,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,..

The value of Tn/( Tn - 1) approximates to 1/(Phi2 - 1) = Phi2.

Thus the value of Tn/( Tn - 1) e.g. 89/55 term approximates to Phi2.

This can be expressed by saying that the nth term in the series divided by the sum of previous one term = 1/(Phi2 - 1).
 
 

However this is the result of a more universal phenomenon.

The 3-term case is

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, .

The value of the Tn/{ ( Tn - 1) + ( Tn - 2) } = 1/(Phi3 - 1) where n approaches infinity

Thus when n = 12,

Tn = 149; Tn - 1 + Tn - 2 = 81 + 44 = 125

Thus Tn/{( Tn - 1) + ( Tn - 2)} = 149/ 125 approximates 1/(1.839286755. - 1)
 
 

The 4-term case is

0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,

Tn/{ ( Tn - 1) + ( Tn - 2) + ( Tn - 3) } = 1/(Phi4 - 1)

Thus when n = 14,

Tn = 401; ( Tn - 1) + ( Tn - 2) + ( Tn - 3) = 208 + 108 + 56 = 372

Thus Tn/{ ( Tn - 1) + ( Tn - 2) + ( Tn - 3) } = 401/ 372 = 1/(1.9275619. - 1)
 
 

Thus in the k-term case

Tn/{ ( Tn - 1) + ( Tn - 2) + ( Tn - 3) +..+ ( Tn - k + 1) } = 1/(Phik - 1)
 
 

These results can be extended to other cases

For example in the 2-term Pell series where a = 2, b = 1 (i.e. where we combine multiples of twice the last term with the previous term to obtain the next term) the following series emerges.

0, 1, 2, 5, 12, 29, 70, 169,

R21 = 2.4142..
 
 

Thus when n= 8

Tn/( Tn - 1) = 169/70 which approximates 1/(R21 - 2)
 
 

Thus in the k-term Pell case where twice the first is combined with each of the other k - 1 terms

Tn/{ ( Tn - 1) + ( Tn - 2) + ( Tn - 3) +..+ ( Tn - k + 1) = 1/(R2111k-1 - 2)
 
 

Thus in the more general case where m times the first term is combined with each of the other k - 1 terms

Tn/{ ( Tn - 1) + ( Tn - 2) + ( Tn - 3) +..+ ( Tn - k + 1) = 1/(Rm111k-1 - k)