Is $\theta=\bar{x}$ a sufficient estimator of $f(x)=\dfrac{1}{\theta}\left(1-\dfrac{1}{\theta}\right)^{x-1}, \ x=1,2,3...$?

Question

My attempt:

The likelihood function is

$$L(\theta)=f(x_1,x_2,...x_n; \theta)=\left(\dfrac{1}{\theta-1}\right)^n\left(\dfrac{\theta -1}{\theta}\right)^{n \bar{x}}$$

Where

$$\bar{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$$

$L$ cannot be factored as

$$g(\theta, \bar{x})h(x_1,x_2,...,x_n)$$

then by Fisher Neymann theorem $\bar{x}$ is not a sufficient estimator.

Is my attempt correct?