I' would like to know whether or not the following sentences are correct and why:
I have been sick since last week.
I have been sick since the last week.
I have been sick for the last week.
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I have been sick since the last week is ungrammatical.
The choice between the other two depends on the context. I have been sick since last week emphasises that you started being sick one week ago and that you are still sick. I have been sick for the last week indicates that your sickness is of at least seven days duration, but it is vaguer about when your sickness started. It could have started a week ago, but it doesn’t exclude the possibility that you were feeling unwell before.
They're all correct (to a degree), depending on the intended meaning:
"I have been sick since last week." That could mean the sickness began last week. It could even mean that the sickness began after last week, if the word "sick" were emphasized.
"I have been sick since the last week " That is incorrect on its own, but could be correct if, say, " of term." were added, meaning the sickness began in the last week of term and is still ongoing.
"I have been sick for the last week." That suggests an ongoing process. At a stretch, it could even mean that the status, "sick all week", has occurred for the last time and "it won't happen again".
Certainly, to convey a specific meaning, some punctuation, emphasis or reordering would greatly help, especially if being shown out of context.