The image shows a sequence of patterns made of triangles, and the task is to determine the rule for the number of triangles in the $n$th pattern.

Looking at the patterns:

• Pattern 1 has 3 triangles.
• Pattern 2 has 6 triangles.
• Pattern 3 has 9 triangles.

It appears that each pattern adds 3 triangles. Thus, the relationship between the pattern number $n$ and the number of triangles in that pattern is linear. Specifically, the number of triangles in the $n$th pattern is $3n$, because for each pattern, 3 triangles are added.

Therefore, the number that should go in the box is 3.

The rule for the number of triangles in the $n$th pattern is:

$\text{Number of triangles} = 3n$

Would you like further details or have any other questions?

Here are 5 related questions:

1. How can you generalize this rule to other sequences where a constant number is added?
2. How would the pattern change if 4 triangles were added instead of 3?
3. What is the 5th pattern in this sequence, based on the rule?
4. How would you describe this sequence using an arithmetic progression?
5. Can you derive a formula for the sum of triangles up to the $n$th pattern?

Tip: Whenever you see a sequence with a constant increase, it's often a linear function, and you can identify the rule by determining the rate of change.