### Instruction: For the arithmetic sequence $1000, ~987, ~974, ~961, \ldots ,$ what is the least positive integer in the sequence?### Response: To find the least positive integer in the sequence, I need to know how the sequence is changing from term to term.
I can see that the difference between consecutive terms is $-13$, so the sequence is decreasing by $13$ each time.
This means that the general term of the sequence is $a_n = 1000 - 13(n-1)$, where $n$ is the position of the term in the sequence.
To find the least positive integer, I need to find the largest value of $n$ such that $a_n > 0$.
I can rewrite the inequality as $1000 - 13n + 13 > 0$, or $13n < 1013$.
Dividing both sides by $13$, I get $n < 77.92$.
Since $n$ has to be an integer, the largest possible value of $n$ is $77$.
This means that the least positive integer in the sequence is $a_{77}$.