### Instruction: Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$?### Response: I need to find all the positive integers that have four digits in base $2$, and then add them up.
The smallest four-digit number in base $2$ is $1000_2$, which is equal to $8$ in base $10$.
The largest four-digit number in base $2$ is $1111_2$, which is equal to $15$ in base $10$.
So $S$ contains all the numbers from $8$ to $15$ in base $10$, or from $1000_2$ to $1111_2$ in base $2$.
There are $8$ numbers in $S$, and they form an arithmetic sequence with a common difference of $1$ in base $10$.
To find the sum of an arithmetic sequence, I can use the formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.
In this case, $n = 8$, $a_1 = 8$, and $a_n = 15$.
So the sum of $S$ in base $10$ is $S_8 = \frac{8}{2}(8 + 15) = 4 \times 23 = 92$.
Now I need to convert $92$ to base $2$.
I can use repeated division by $2$ and write the remainders from bottom to top.
$92 \div 2 = 46$ with a remainder of $0$.
$46 \div 2 = 23$ with a remainder of $0$.
$23 \div 2 = 11$ with a remainder of $1$.
$11 \div 2 = 5$ with a remainder of $1$.
$5 \div 2 = 2$ with a remainder of $1$.
$2 \div 2 = 1$ with a remainder of $0$.
$1 \div 2 = 0$ with a remainder of $1$.
So $92_{10} = 1011100_2$.