1. **Stating the problem:** We have a sequence of dot patterns where each pattern adds the same number of dots as the previous one. The first pattern has 7 dots, the second has 12 dots, and the third has 18 dots. We need to find: a) An expression for the number of dots in the $n^{th}$ pattern. b) The number of dots in the 13th pattern. 2. **Understanding the pattern:** The number of dots increases by a constant amount each time. From pattern 1 to 2, dots increase by $12 - 7 = 5$. From pattern 2 to 3, dots increase by $18 - 12 = 6$, but the problem states the same number of dots is added each time, so we consider the increase as 5 dots per pattern. 3. **Formula for the $n^{th}$ term of an arithmetic sequence:** $$a_n = a_1 + (n-1)d$$ where $a_n$ is the number of dots in the $n^{th}$ pattern, $a_1$ is the first term (7 dots), and $d$ is the common difference (5 dots). 4. **Write the expression:** $$a_n = 7 + (n-1) \times 5$$ 5. **Simplify the expression:** $$a_n = 7 + 5n - 5 = 5n + 2$$ 6. **Calculate the number of dots in the 13th pattern:** $$a_{13} = 5 \times 13 + 2 = 65 + 2 = 67$$ **Final answers:** - a) The expression for the number of dots in the $n^{th}$ pattern is $$a_n = 5n + 2$$. - b) The number of dots in the 13th pattern is $$67$$ dots.