For which the below fraction refer to?

Question

I have got this fraction representation :$$a=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots}}}} $$ but i can't know for which it's refer to , I mixed that with Golden ratio however the fractionof Golden ratio is started with 1+ and that started with 0 , Now how i can get the correct Algebraic equation which refer to that number and what is it ?

Answer 1

You can write your expression as $$a=\frac{1}{1+a}$$ which gives the quadratic $a^2+a-1=0$, and so $$a=\frac{-1\pm\sqrt 5}{2}$$