Solve the equation `1+1/(1+1/(1+1/(1+1/(1+1/x))))=x` When in expression on left hand side the sign of a fraction is repeated `n` times.

Explanation

Given equation is `1+1/(1+1/(1+1/(1+1/(1+1/x)))))=x`
Let us replace `x` on the LHS of the given equation by the expression of `x`. This result in an equatioin of the same form, which however involves `2n` fraction lines. Continuing this process on the basis of this transformation, we can write
`x=1+lim_(ntooo)1+1/(1+1/(1+1/(1+1/(1+1/x)`[ `n` fractions]
`impliesx+1+1/ximpliesx^(2)=x-1=0`
`:.x=(1+1sqrt(5))/2`
`:.x_(1)=(1+sqrt(5))/2,x_(2)=(1-sqrt(5))/2`
Satisfy the given equation and this equation has no other roots.