1. **Stating the problem:** We have a sequence of dot patterns where each pattern adds the same number of dots compared to the previous one. We want to find an expression for the number of dots in the $n^{th}$ pattern. 2. **Observing the pattern:** - For $n=1$, there are 5 blue dots arranged vertically. - For $n=2$, the original 5 blue dots remain, and 3 purple dots are added. - For $n=3$, the original 5 blue dots remain, and 2 groups of 3 purple dots each are added (total 6 purple dots). 3. **Identifying the pattern in dots added:** - The first pattern has 5 dots. - Each subsequent pattern adds 3 dots more than the previous pattern. 4. **Formulating the expression:** - The number of dots in the $n^{th}$ pattern is the initial 5 dots plus $3$ dots added for each pattern after the first. - This can be written as: $$\text{dots}_n = 5 + 3(n - 1)$$ 5. **Simplifying the expression:** $$\text{dots}_n = 5 + 3n - 3 = 3n + 2$$ 6. **Explanation:** - The term $3n$ accounts for the 3 dots added per pattern number. - The $+2$ adjusts for the initial 5 dots in the first pattern. **Final answer:** $$\boxed{\text{dots}_n = 3n + 2}$$ This formula gives the total number of dots in the $n^{th}$ pattern.