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

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

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Answer 1 · 2018-08-06T16:09:37

This is a flash, from my memory.

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.