A remarkably simple (circular) type equation shows the general value for phi_n.

(Phi_n)ⁿ = 1/(2 - phi_n)

Thus when n = 1,

Phi₁ = 1/(2 - phi₁). Thus here, phi₁ = 1.
 
 

When n = 2,

(Phi₂)² = 1/(2 - phi₂). Apart from the trivial solution i.e. 1 (which satisfies the equation for all values of n), the real valued solution for phi₂ (i.e. phi) is 1.6180339887..,
 
 

When n = 3,

  (Phi₃)³ = 1/(2 - phi₃).

As we have seen the value for phi₃ = 1.839286755…
 
 

When n = 4,

(Phi₄)⁴ = 1/(2 - phi₄).

The value here for phi₄ = 1.9275619…
 
 

So it goes for any value of n.
 
 
 
 

In terms of connections as between the different ratios, a fascinating approximating type relationship exists.

In general terms it can be expressed as
 
 

Phi_n/Phi_2n = Phi_{n +1} - 1

Thus when n = 1, we get Phi₁/Phi₂ =1/1.6180339887… = .6180339887…

= Phi₁₊₁ - 1 (exactly in this case).
 
 

When n = 2, we get Phi₂/Phi₄ = 1.6180339887…/1.9275619..

= .83942.

This reasonably approximates the value of Phi_{2 +1} - 1, i.e. Phi₃ - 1 = .839286755...
 
 

When n= 3, we get Phi₃/Phi₆ = 1.839286755../1.98358..

= .92726

This reasonably approximates the value of Phi_{3 +1} - 1 i.e. Phi₄ - 1 = .9275619
 
 

With n = 4, we get Phi₄/Phi₈ = 1.9275619../ 1.99604..

= .96569

This reasonably approximates the value of Phi_{4 +1} - 1, i.e. Phi₅ - 1

= .96594..
 
 

With n = 5, we get Phi₅/Phi₁₀ = 1.9659482../ 1.99902

= .98345..

This reasonably approximates the value of Phi_{5 +1} - 1, i.e. Phi₆ - 1

= .98358..
 
 

Finally - to illustrate - with n = 6, we get Phi₆/Phi₁₂ = 1.98358../ 1.999756

= .99191..

Once again, this reasonably approximates the value of Phi_{6 +1} - 1, i.e. Phi₇ - 1

.9919
 

So the approximation tends to improve for larger values of n.
 
 
 
 

I will give one more example here of a precise and remarkable connection as between phi (i.e. phi₂) and phi_n.

(Phi_n - 1)/(Phi₂ - 1) - 1 divided by (Phi_n/Phi₂) - 1 = Phi₂² (for n > 2).

Thus when n = 3,

(Phi_n - 1)/(Phi₂ - 1) - 1 = (1.839286755. - 1)/(1.6180339887. - 1) - 1 = (.839286755./.6180339887) - 1 = .357994.

(Phi_n/Phi₂) - 1 = (1.839286755.)/(1.6180339887.) - 1

= .1367417.

Dividing both results we get .357994/.13674 = 2.618034 = Phi₂².
 
 

For example when n = 5,

(Phi_n - 1)/(Phi₂ - 1) - 1 = (1.9659482. - 1)/(1.6180339887. - 1) - 1 =

(.9659482./.6180339887.) - 1 = .562937.

(Phi_n/Phi₂) - 1 = (1.9659482..)/(1.6180339887.) - 1

= .2150228.

Dividing both results we get .562937/.2150228 = 2.618033..= Phi₂²
 
 

Another remarkable general result can also be given as follows
 
 

(Phi_n² - 1)/(Phi₂² - 1) - 1 divided by (Phi_n²/Phi₂²) - 1 = Phi₂

For example when n = 3,

(Phi_n² - 1)/(Phi₂² - 1) - 1 = (3.382975767. - 1)/(2.6180339887 - 1) - 1

= (2.382975767/1.6180338887) - 1 = .47276..
 
 

(Phi_n²/Phi₂²) - 1 = (3.382975767.. )/(2.6180339887..) - 1

= .292182

Dividing both results we get .47276/.292182 = 1.618033 = Phi₂
 
 

Similar - though less elegant - general results can be found for powers of Phi_n and Phi₂ >2.