In the 2-term Fibonacci case,

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

The value of T_n/( T_{n - 1}) approximates to 1/(Phi₂ - 1) = Phi₂.

Thus the value of T_n/( T_{n - 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 T_n/{ ( T_{n - 1}) + ( T_{n - 2}) } = 1/(Phi₃ - 1) where n approaches infinity

Thus when n = 12,

T_n = 149; T_{n - 1} + T_{n - 2} = 81 + 44 = 125

Thus T_n/{( T_{n - 1}) + ( T_{n - 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,…

T_n/{ ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) } = 1/(Phi₄ - 1)

Thus when n = 14,

T_n = 401; ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) = 208 + 108 + 56 = 372

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

Thus in the k-term case

T_n/{ ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) +…..+ ( T_{n - k + 1}) } = 1/(Phi_k - 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

T_n/( T_{n - 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

T_n/{ ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) +…..+ ( T_{n - k + 1}) = 1/(R_2111…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

T_n/{ ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) +…..+ ( T_{n - k + 1}) = 1/(R_m111…k-1 - k)