I was wondering whether there is a universal formula to find the closed form of a recursive compound geometric sequence. With $d$ being the default value, $a$ being the added value each time, and $n$ being the exponent raised.
$$ f(x)= \begin{cases} x=1:d \\ x>1:(a+f(x-1))^n \end{cases} $$ For example, if $x=3$, $n=\frac{1}{2}$, $a=2$, and $d=\phi$, we get $$ \sqrt{2+\sqrt{2+\phi}}$$ Every time $x$ increases, we add a square root. $$x=4:\sqrt{2+\sqrt{2+\sqrt{2+\phi}}} \\ x=5:\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\phi}}}} \\ \cdots$$