Complete the Arithmetic Sequence: 10, -10, -30, ?

To solve this problem, we'll follow these steps:

• Step 1: Calculate the difference between the first and second terms.
• Step 2: Calculate the difference between the second and third terms.
• Step 3: Identify a pattern from these differences.
• Step 4: Apply the pattern to determine the fourth term.

Now, let's work through each step:

Step 1: Calculate the difference between the first and second terms $10$ and $-10$:

The calculation is: $10 - (-10) = 10 + 10 = 20$.

Step 2: Calculate the difference between the second term $-10$ and the third term $-30$:

The calculation is: $-10 - (-30) = -10 + 30 = 20$.

Step 3: Notice the pattern; each difference is $20$. The series decreases by $20$ each time.

Step 4: Apply this consistency to determine the next number in the sequence:

Calculate the difference from the third number $-30$:
$-30 - 20 = -50$.

Therefore, the missing number is $\mathbf{-50}$, which corresponds to choice $4$.

Thus, the solution to the problem is $-50$.