Geometric progression

A sequence $ (x_n)_{n\in N} $ $ x_n\ne 0 $ is in geometric progression if and only if

$$x_n^2=x_{n-1}x_{n+1}$$

or

$$\frac{x_{n+1}}{x_n}=\frac{x_n}{x_{n-1}}=\rho$$

so

$$x_1=\rho x_0$$
$$x_1=\rho  x_0$$
$$x_2=\rho  x_1=\rho^2  x_0$$
$$x_n=\rho ^{n-1} x_0$$

The partial sum of geometric progression in case $ \rho\ne 0 $ is

$$s_n=x_0+x_1+...+x_n=(1+\rho+\rho^2+...+\rho^{n-1})x_0=\frac{1-\rho^n}{1-\rho}x_0$$

The sum of geometric progression is , for

$$|\rho|<1\: as \lim_{n\to \infty}\rho ^n=0$$
$$s=\sum_{n=0}^{+\infty}x_n= \frac{1}{1-\rho}x_0$$

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