Question

`A=1+1/{1+1/[1+1/(1+1/...)]}` by solving it i get both a positive and a negative solution. Why negative?

`A=1+1/{1+1/[1+1/(1+1/{1+1/[1+1/(1+1/...)]})]}`

Answer 1

This is a flash, from my memory.

Explanation:

Here, `A = 1 + 1/A` generates the Continued Fraction ( CF ).

There are two A-intercepts, `1/2 ( 1 +- sqrt 5)`, of the hyperbola

B = A- ( 1 + 1/A ).

See graph.
graph{y-x+1+1/x=0}
Inversely,

the two branches of the hyperbola have separate equations

`A = 1/2 ( B + 1 +sqrt( ( B^2 + 2 B + 5 )), A > 0` and

`A = 1/2 ( B + 1 - sqrt( ( B^2 + 2 B + 5 )), A < 0`

The A values under reference are against B = 0.
graph{y-x+1+1/x=0}

The positive intercept is from the branch in #Q_1 and Q_4. The

other is from the opposite branch.