Let's analyze the pattern.

Part (a): Expression for the number of dots in the $n$-th pattern.

By examining the given sequence:

• Pattern 1 has 6 dots.
• Pattern 2 has 9 dots.
• Pattern 3 has 12 dots.

It appears that the number of dots increases by 3 for each successive pattern. This suggests an arithmetic sequence where:

• The first term ($a_1$) = 6.
• The common difference ($d$) = 3.

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Substituting the known values: $a_n = 6 + (n - 1) \cdot 3$ Simplifying: $a_n = 6 + 3n - 3 = 3n + 3$

So, the expression for the number of dots in the $n$-th pattern is: $\boxed{a_n = 3n + 3}$

Part (b): Number of dots in the 14th pattern.

Now, substitute $n = 14$ into the formula: $a_{14} = 3(14) + 3 = 42 + 3 = 45$

So, the 14th pattern has $\boxed{45}$ dots.

Would you like more details or have any questions on this?

5 Related Questions:

1. How do you identify arithmetic sequences in other patterns?
2. Can you explain the steps to derive the formula for an arithmetic sequence?
3. How would the pattern change if the number of dots increased by 4 instead of 3?
4. Can you generalize this formula for any starting number of dots and increment?
5. How can you graph this arithmetic sequence?

Tip:

When working with arithmetic sequences, always identify the first term and the common difference to easily build the general formula.