Let's analyze the sequence of dots in the given pattern.

From the image, it looks like we have:

• Pattern 1: 3 dots
• Pattern 2: 6 dots
• Pattern 3: 9 dots

It seems that each pattern number increases by 3 dots each time. This tells us that the sequence is an arithmetic sequence where each new pattern adds 3 dots.

To find the number of dots in the 10th pattern:

Part (a)

1. Identify the sequence formula. Since this is an arithmetic sequence, we can use the formula: $a_n = a_1 + (n - 1) \cdot d$ where:

   • $a_n$ is the number of dots in the $n$-th pattern,
   • $a_1$ is the number of dots in the first pattern (3),
   • $d$ is the common difference (3).

2. Substitute the values: $a_{10} = 3 + (10 - 1) \cdot 3$ $a_{10} = 3 + 9 \cdot 3$ $a_{10} = 3 + 27 = 30$

So, there are 30 dots in the 10th pattern.

Part (b)

To explain how this was calculated, we observe that each pattern adds the same number of dots (3), forming an arithmetic sequence. Using the formula for an arithmetic sequence, we calculated the 10th term by starting from 3 dots in the first pattern and adding 3 dots for each subsequent pattern up to the 10th.

Would you like further clarification or details on any specific part?

Here are five related questions to explore further:

1. How many dots would be in the 15th pattern?
2. What would the total number of dots be if we added up all patterns from the 1st to the 10th?
3. Could you identify the common difference in an arithmetic sequence, and how does it affect the sequence?
4. How would you describe the sequence if each pattern added 5 dots instead of 3?
5. How can we represent this arithmetic sequence graphically?

Tip: In an arithmetic sequence, the difference between consecutive terms is constant, making it easy to find any term with the formula $a_n = a_1 + (n - 1) \cdot d$.