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/(R2111…k-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/(Rm111…k-1 - k)