An arithmetic sequence has first term 2018 and common difference -5. From which term onwards does the sequence become negative?

Step-by-step solution

1. 1

   Write the general term. $u_n=2018+(n-1)(-5)=2023-5n$ The sequence decreases by $5$ each step, so once it turns negative it stays negative — there is a single crossing point.

2. 2

   Set up the inequality for negativity. $2023-5n<0\quad\Longrightarrow\quad 5n>2023\quad\Longrightarrow\quad n>404.6$

3. 3

   Take the smallest integer that works. Since $n$ must be a positive integer, the first index satisfying the inequality is $n=405$.

4. 4

   Evaluate the two terms around the crossing. $u_{404}=2023-2020=3>0,\qquad u_{405}=2023-2025=-2<0$

5. 5

   Conclude. $u_{404}$ is the last positive term and $u_{405}=-2$ is the first negative one, so the answer is C.