I was online looking for an apartment. It says, "we'll only need first, last, and deposit once your application is approved."
What does this mean?
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This means that once your application is approved, you'll need to pay your first and last month's rent and a security deposit. You may get your security deposit back when you terminate your lease, assuming you've met all the conditions in the lease and haven't damaged your apartment.
If you decide to extend your lease, your last month's rent is usually carried forward and applied to the last month of your new/extended lease. For example, if you lease an apartment from September 2013 through the end of September 2014, your last month's rent is assumed to apply to September 2014. If you choose to extend your lease through the end of September 2015, your last month's rent is assumed to apply to September 2015, and so on. In other words, your last month's rent applies to the last month of your lease as it's currently stated. If you change/extend the lease, the last month changes.
If your rent increases sometime over the course of your lease (or your lease extension, specifically), you'll usually owe the difference between the amount you paid for your last month and the current rent. For example, if you pay last month's rent of $1500 when you sign your lease, but the rent increases to $1800 when you extend your lease, you'll probably owe $300 in the last month.