### Instruction: Let \[A = \lceil \log_2 2 \rceil + \lceil \log_2 3 \rceil + \dots + \lceil \log_2 1000 \rceil\]and \[B =  \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \dots + \lfloor \log_2 1000 \rfloor.\]Compute $A-B.$.### Response: The first thing I notice is that the ceiling function always rounds up and the floor function always rounds down, so for any integer $n$, \[\lceil \log_2 n \rceil - \lfloor \log_2 n \rfloor\]is either 0 or 1, depending on whether $n$ is a power of 2 or not.
This means that the difference $A-B$ is equal to the number of powers of 2 between 2 and 1000, inclusive.
To find this, I need to think about how many times I can double 2 until I reach or exceed 1000.
I know that $2^9 = 512$ and $2^{10} = 1024$, so $2^9$ is the largest power of 2 that is less than or equal to 1000.
Therefore, there are 9 powers of 2 between 2 and 1000, namely $2, 2^2, 2^3, \dots, 2^9$.
So the final answer is 9.