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When I click the Users button in the left navigation bar, I am taken to a page that lists users by reputation increase. What seems surprising is that, many of the users listed don't seem to have posted any answers or questions recently. When I check the Activity page on their profiles, it shows that their most recent answers, for example, were posted years ago.

So, how is it that these mostly inactive users are gaining so much reputation on a yearly or monthly or even weekly basis?

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In general, it's not surprising that people who were active on the site years ago are getting the passive reputation increases observed.

We get reputation not when posting something, but when someone upvotes it. A post can be upvoted any time after it is posted (unless it's locked or deleted), and posts that get a lot of views continue to get upvotes over time, thus giving their authors reputation points (unless the post is a community wiki).

To see upvotes over time, you can look at the timeline of a post. You can access this by clicking the little grey clock icon under a post's score. As an example, take a look at the timeline of the most upvoted question on Ask Ubuntu. You can see that this question has received upvotes every few days since it was posted. Check out the reputation tab of the author's profile and you can see the rep from just that question accruing.

When a question has more views, it gets even more views as it appears higher in search engine results. So, the posts that get more views and more upvotes are generally old. Also, most of the questions people have about common problems and features were answered a long time ago and are still getting views. So, the old posts with many views, asked or answered years ago by folks who are now inactive here or have stopped asking or answering questions to focus on whatever else they feel like doing, are getting more upvotes than the vast majority of new questions posted today.

So, the trend of inactive users who answered many questions in the past getting bigger reputation increases than many people active today is an expected effect of the way voting and reputation work.

Personally, I find it very interesting to see which of my answers continue to get upvotes over time, as this goes some way to indicating which ones were actually the most useful. It's often the case that the answers that get the most attention around the time of posting don't receive any later, while answers that were barely noticed gradually get upvoted more and more over the years. An even better way to see this is to check the low rep/unregistered feedback on your posts, for example using this SEDE query.

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