In the2-termFibonacci case,

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

Tapproximates to_{n}/( T_{n - 1})1/(Phi._{2}- 1) = Phi_{2}Thus the value of T

_{n}/( T_{n - 1}) e.g. 89/55 term approximates toPhi._{2}This can be expressed by saying that the nth term in the series divided by the sum of previous one term =

1/(Phi._{2}- 1)

However this is the result of a more universal phenomenon.

The

3-termcase 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/(Phiwhere n approaches infinity_{3}- 1)Thus when

n = 12,

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

Tapproximates_{n}/{( T_{n - 1}) + ( T_{n - 2})} = 149/ 1251/(1.839286755. - 1)

The

4-termcase 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/(Phi_{4}- 1)Thus when n = 14,

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

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

Thus in the

k-termcase

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-termPell series wherea = 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, …

R_{21 }= 2.4142..

Thus when

n= 8

Twhich approximates_{n}/( T_{n - 1}) = 169/701/(R_{21}- 2)

Thus in the

k-termPell case where twice the first is combined with each of the otherk - 1terms

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

mtimes the first term is combined with each of the otherk - 1terms

T_{n}/{ ( T_{n - 1}) + ( T_{n - 2}) + ( T_{n - 3}) +…..+ ( T_{n - k + 1}) = 1/(R_{m111…k-1}- k)