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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:

enter image description here

I am also given the following problem :

enter image description here

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?

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

As a side remark, note the typo in the quote where "the second inequality" should be "the second equality".

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