How to understand this property of Geometric Distribution

[christang_1023]

There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$.
I understand it in such a way: $X$ is independent, that's to say after there are $(n+k-1)$ successive failures, $k$ additional trials performed afterward won't be impacted, so these $k$ trials can be treated as isolated trials.
Am I right?

 

[member 587159]

There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$.
I understand it in such a way: $X$ is independent, that's to say after there are $(n+k-1)$ successive failures, $k$ additional trials performed afterward won't be impacted, so these $k$ trials can be treated as isolated trials.
Am I right?

Yes, that's the intuition to remember it. But be sure to be able to prove it using the definition of geometric distribution.