### Instruction: The average of the numbers $1, 2, 3,\dots, 98, 99,$ and $x$ is $100x$. What is $x$?### Response: I need to find the value of $x$ that makes the average of these 100 numbers equal to $100x$.
The average of a set of numbers is the sum of the numbers divided by the number of numbers, so I can write an equation for the average: $\frac{1 + 2 + 3 + \cdots + 98 + 99 + x}{100} = 100x$.
To simplify this equation, I can use the formula for the sum of an arithmetic series.
The first term is $1$, the common difference is $1$, and the last term is $99$, so the sum of the first $99$ terms is $\frac{99}{2}(1 + 99) = 4950$.
Adding $x$ to both sides, I get $\frac{4950 + x}{100} = 100x$.
Multiplying both sides by $100$, I get $4950 + x = 10000x$.
Subtracting $x$ from both sides, I get $4950 = 9999x$.
Dividing both sides by $9999$, I get $x = \frac{4950}{9999}$.