1. **Stating the problem:** We have a sequence of dot patterns where the number of dots increases by the same amount each time. We need to find an expression for the number of dots in the $n^{\text{th}}$ pattern and then find how many dots are in the 13$^{\text{th}}$ pattern. 2. **Observing the pattern:** - Pattern 1 has 5 dots. - Pattern 2 has 10 dots. - Pattern 3 has 15 dots. 3. **Identifying the rule:** The number of dots increases by 5 each time (from 5 to 10 to 15). 4. **Formula for the $n^{\text{th}}$ term of an arithmetic sequence:** $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term and $d$ is the common difference. 5. **Applying values:** - $a_1 = 5$ - $d = 5$ So, $$a_n = 5 + (n-1) \times 5 = 5 + 5n - 5 = 5n$$ 6. **Expression for the number of dots in the $n^{\text{th}}$ pattern:** $$\boxed{a_n = 5n}$$ 7. **Finding the number of dots in the 13$^{\text{th}}$ pattern:** $$a_{13} = 5 \times 13 = 65$$ **Final answer:** - a) Number of dots in the $n^{\text{th}}$ pattern is $5n$. - b) Number of dots in the 13$^{\text{th}}$ pattern is 65.