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 = T_{n + 1} (phi₂)² - (T_{n + 1} - T_n)
 

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 + phi_2, 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 phi_kⁿ 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.