Which one is correct?
- This week is the final week of my staying in this city.
- This week is the last week of my staying in this city.
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The words Final and Last are synonyms. They can be used interchangeably most of the time. But it doesn't mean that both of these words imply the same meaning in all contexts. The context plays a vital role on whether to use Final or Last in a sentence. The word 'Final' implies the only being remaining or ultimate. So, "the final week of staying in this city" suggests that there will be no more weeks for staying in this city or it is the only being remaining week for staying in this city. On the other hand, the word "Last" means coming at the end of a series. "The last week for staying in this city" simply means that this week is the last week of all other previous weeks for staying in this city.
Furthermore, the word 'Final' is more formal than 'Last'. To cut the long story short, Final means 'Last' according to http://dictionary.cambridge.org/dictionary/english/final , Therefore, the former statement implies the letter statement without having much difference between them.
They are both correct. You can also emphasize your emotion like: This is the final week of my staying in this city, I am glad it is over. Or: This is the last week of my staying in this city, I wish it could have been a longer stay
Others have pointed out the lack of difference in the use of final/last for this particular pair of sentences, but I think it should be pointed out that there is a difference in other contexts.
For example, there's a very common use of "last week" which cannot be replaced by "final week":
Last week I stayed in the city.
In the last week, I stayed in three hotels.
Here, "last" is being used to mean "previous". In these contexts, last and final are not interchangeable, as "final" has an absolute sense, and these sentence do not imply anything about what will happen next week. (e.g. You could continue to stay in the city, or stay in additional hotels.)
The usage can be highly context dependent, as it's somewhat easy to change the sense between the two. For example, adding "of my journey" to the second sentence changes things from a relative to an absolute sense, meaning that "final" can now be substituted:
In the last week of my journey, I stayed in three hotels.
In the final week of my journey, I stayed in three hotels.
As was pointed out in the comments even the phrase "In the last week, I stayed in three hotels." can differ in whether "final" is an appropriate substitution, depending on the precise context. If it's interpreted as "In the seven days leading up to now, I stayed in three hotels", then "last" and "final" are not interchangeable. However, if the context is something like "I spent four weeks in Europe. In the first three weeks I stayed with a friend, and then for the remainder I stayed in three hotels.", then "last" and "final" are interchangeable.