### 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: I know that $\log_2 2 = 1$, so $\lceil \log_2 2 \rceil = \lfloor \log_2 2 \rfloor = 1$.
Similarly, $\log_2 4 = 2$, so $\lceil \log_2 4 \rceil = \lfloor \log_2 4 \rfloor = 2$.
In general, for any integer power of 2, say $2^n$, we have $\lceil \log_2 2^n \rceil = \lfloor \log_2 2^n \rfloor = n$.
However, for any integer that is not a power of 2, say $m$, we have $\lceil \log_2 m \rceil = \lfloor \log_2 m \rfloor + 1$.
This is because $\log_2 m$ is not an integer, so $\lceil \log_2 m \rceil$ is the smallest integer that is greater than $\log_2 m$, and $\lfloor \log_2 m \rfloor$ is the largest integer that is less than $\log_2 m$.
Therefore, the difference $\lceil \log_2 m \rceil - \lfloor \log_2 m \rfloor$ is always 1 for non-power-of-2 integers.
Now, I want to count how many terms in the sums $A$ and $B$ are equal, and how many are different by 1.
I notice that the powers of 2 in the range from 2 to 1000 are 2, 4, 8, 16, 32, 64, 128, 256, 512.
There are 9 of them, so there are 9 terms in $A$ and $B$ that are equal.
The rest of the terms, which are 991 of them, are different by 1.
So, to find $A-B$, I just need to add up the differences of all the terms.
Since the difference is always 1, I just need to count how many terms are different.
That is 991, so $A-B = 991$.