Addition and Subtraction of Directed Numbers: Regularity

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

• Step 1: Identify the pattern in the sequence.
• Step 2: Calculate the common difference.
• Step 3: Use the common difference to find the next term in the sequence.

Step 1: Look at the given sequence: $-15, -17, -19, -21$.
To find the common difference, we can subtract each term from the next one. For example, the difference between $-15$ and $-17$ is:

$-17 - (-15) = -17 + 15 = -2$

Similarly, between $-17$ and $-19$, and $-19$ and $-21$, we have:

$-19 - (-17) = -2$

$-21 - (-19) = -2$

Step 2: We have determined that the common difference $d$ is $-2$.

Step 3: To find the next term, subtract the common difference from the last term: $-21 - 2 = -23$.

Therefore, the next number in the sequence is $-23$.

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

• Step 1: Identify the pattern present in the sequence.
  The sequence provided is $1, 0, -1, -2, -3, -4$.
• Step 2: Recognize that each number is decreasing by 1.
  This indicates an arithmetic sequence with a common difference of $-1$.
• Step 3: Determine the next number using the pattern.
  Starting with $-4$, subtract $1$ to find the next term.

Now, let's work through each step:
Step 1: The sequence starts at $1$ and each subsequent number decreases by $1$.
Step 2: Identify this consistent decrease signifying an arithmetic sequence with a common difference $d = -1$.
Step 3: Calculate the next term after $-4$:
$-4 - 1 = -5$

Therefore, the next number in the sequence is $-5$.

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

• Step 1: Confirm the pattern in the sequence by calculating differences between terms.
• Step 2: Calculate the next term in the sequence using the identified pattern.

Let's analyze the sequence $-16, -22, -28, -34$:

Step 1: Calculate the difference between consecutive terms:
$-22 - (-16) = -6$
$-28 - (-22) = -6$
$-34 - (-28) = -6$
It is clear that the common difference, denoted as $d$, is $-6$.

Step 2: Add the common difference to the last term to find the next number:
The last number is $-34$. Add the common difference $d = -6$ to find the next term:
$-34 + (-6) = -40$

Therefore, the next number in this sequence is $-40$.

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 $-50$.