Calculating Jeffreys Prior for geometric distribution

This question is already answered here, but I would like to know why it is worked out the way it is

My lecture notes state the following:

I am also given the following problem :

Now, what I thought needs to be done is that we need to first find the joint likelihood, i.e. $$f(X|\theta)=\prod_{i=1}^n (1-\theta)^{x_i-1}\theta$$ But, the answer in my notes, as well as the one in the hyperlink I stated at the top don't bother finding it and they work with the likelihood of one observation. Can someone explain to me why?

The information brought by $$n$$ iid observations is $$n$$ times the information brought by one observation. They both lead to the same Jeffreys prior.