What is the sum of the integers from $-30$ to $50$, inclusive?
To find the sum of a sequence of integers, I can use the formula for the sum of an arithmetic series. An arithmetic series is a sequence of numbers with a constant difference between consecutive terms. The first term of the series is $-30$, the last term is $50$, and the common difference is $1$. The formula for the sum of an arithmetic series is $S_n = \frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. To find $n$, I can use the formula for the nth term of an arithmetic series, which is $a_n = a_1 + (n-1)d$. Plugging in the values, I get $50 = -30 + (n-1)1$. Simplifying, I get $80 = n - 1$, so $n = 81$. Now I can plug in $n$, $a_1$, and $a_n$ into the formula for the sum. I get $S_n = \frac{81}{2}(-30 + 50)$. Simplifying, I get $S_n = \frac{81}{2}(20) = 810$. 