How to iterate the geometric mean

Question

I've been looking for the formula of the rolling geometric mean, which could be calculated iteratively, i.e $G(x)=f(G(x-1))$. This is needed for the computationally efficient spreadsheets.

Answer 1

$$G_{n} = \bigg(\prod_{i=1}^n x_i\bigg)^{1/n} = \bigg(x_n\prod_{i=1}^{n-1} x_i\bigg)^{1/n} = x_n^{1/n}\bigg(\prod_{i=1}^{n-1} x_i\bigg)^{1/n} = \left(x_n\left(G_{n-1}\right)^{n-1}\right)^{\frac 1n}$$