To determine the missing number in the sequence $4, 7, 12, \_\_, 37$, we need to identify a pattern.

Let's examine the differences between consecutive terms:

$7 - 4 = 3$ $12 - 7 = 5$ $\text{Next difference (missing number) } = x - 12$ $37 - x = \text{Final difference}$

Notice that the differences are increasing by 2:

• $3$ is the first difference,
• $5$ is the second difference.

If this pattern continues, the next difference should be $7$.

So, the next term in the sequence is:

$x = 12 + 7 = 19$

Let's verify by checking the final difference:

37 - 19 = 18 \quad \text{(which is 19 - 12 = 7 + 2 \times 5)}

The pattern is consistent. Thus, the missing number is 19.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is an arithmetic sequence?
2. How can you identify patterns in sequences?
3. What is the general formula for the nth term of an arithmetic sequence?
4. What happens if the differences between terms are not consistent?
5. How can you identify other types of sequences, like geometric sequences?

Tip: Always check if the differences between terms are consistent when identifying a pattern in sequences.