IMPLICATION OF EXPANSION FOR 2-TERM SEQUENCE

The formulas I have mentioned in the previous two articles can be extended to the two-term Fibonacci sequence. Once again this series is

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

In this case the relevant general formula is

(phi2)n = Tn + 1 (phi2)2 - (Tn + 1 - Tn)

Thus when n = 1,

phi = (phi2)2 - 1

When n = 2,

(phi2)2 = (phi2)2

When n = 3,

(phi2)3 = 2(phi2)2 - 1

When n = 4,

(phi2)4= 3(phi2)2 - 1

When n = 5,

(phi2)5 = 5(phi2)2 - 2

When n = 6,

(phi2)6 = 8(phi2)2 - 3

When n = 7,

(phi2)7 = 13(phi2)2 - 5

When n = 8,

(phi2)8 = 21(phi2)2 - 8

and so on

As (phi2)2 = 1 + phi2, this formulation reduces to the following well-known expressions (already mentioned in the previous two articles).

When n = 1,

phi2 = phi2

When n = 2,

(phi2)2 = 1 + phi2

When n = 3,

(phi2)3 = 2(phi2)+ 1

When n = 4,

(phi2)4= 3(phi2)+ 2

When n = 5,

(phi2)5 = 5(phi2)+ 3

When n = 6,

(phi2)6 = 8(phi2)+ 5

When n = 7,

(phi2)7 = 13(phi2)+ 8

When n = 8,

(phi2)8 = 21(phi2)+ 13

etc.

So not alone are the expressions for phikn governed in each case for the 2-term, 3-term and 4-term (i.e. k = 2, 3 and 4 respectively) but a general formula itself governs the expressions in all cases.

Other expansions for phi2n are also possible

phi2 = 1/phi2 + 1

Therefore,

When n = 2,

(phi2)2 = 1/phi2 + 2

When n = 3,

(phi2)3 = 2/phi2+ 3

When n = 4,

(phi2)4= 3/phi2+ 5

When n = 5,

(phi2)5 = 5/phi2+ 8

When n = 6,

(phi2)6 = 8/phi2+ 13

When n = 7,

(phi2)7 = 13/phi2+ 21

When n = 8,

(phi2)8 = 21/phi2+ 34

etc.