### Instruction: What is the sum of all the odd integers between $300$ and $500$?### Response: I notice that the odd integers between $300$ and $500$ form an arithmetic sequence, with a common difference of $2$.
To find the sum of an arithmetic sequence, I need to know the first term, the last term, and the number of terms.
The first term is the smallest odd integer greater than $300$, which is $301$.
The last term is the largest odd integer less than $500$, which is $499$.
To find the number of terms, I can use the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $a_n$ is the nth term, and $d$ is the common difference.
Plugging in the values I know, I get $499 = 301 + (n-1)2$.
Simplifying, I get $198 = (n-1)2$, or $99 = n-1$.
Solving for $n$, I get $n = 100$.
So there are $100$ odd integers between $300$ and $500$.
Now I can use the formula for the sum of an arithmetic sequence: $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.
Plugging in the values I know, I get $S_{100} = \frac{100}{2}(301 + 499)$.
Simplifying, I get $S_{100} = 50 \times 800$.