1. **State the problem:** We have a sequence of dot patterns where each pattern adds the same number of dots as the previous one. We want to find: a) An expression for the number of dots in the $n^{th}$ pattern. b) The number of dots in the 13$^{th}$ pattern. 2. **Analyze the given patterns:** Pattern 1 has 5 dots. Pattern 2 has 7 dots. Pattern 3 has 12 dots. 3. **Find the differences between patterns:** From pattern 1 to 2: $7 - 5 = 2$ From pattern 2 to 3: $12 - 7 = 5$ The difference is not constant, so the sequence is not arithmetic. 4. **Check if the problem states "the same number of dots is added each time":** The problem states the same number of dots is added each time, but the given counts do not reflect that. Possibly the counts in the problem description are approximate or the pattern is misunderstood. 5. **Re-examine the problem's dot counts:** The problem states pattern 1 has 5 dots, pattern 2 has 7 dots, and pattern 3 has 12 dots. Since the problem says the same number of dots is added each time, let's assume the difference is constant and the given counts are correct. 6. **Calculate the common difference:** Let the common difference be $d$. Then pattern 1: $a_1 = 5$ Pattern 2: $a_2 = a_1 + d = 5 + d = 7$ so $d = 2$ Pattern 3: $a_3 = a_2 + d = 7 + 2 = 9$ but given is 12, so this contradicts. 7. **Conclusion:** The problem's description and counts are inconsistent. Since the problem states "the same number of dots is added each time," we will use the first two patterns to find $d=2$ and ignore the third pattern's count. 8. **Write the expression for the $n^{th}$ pattern:** The number of dots in the $n^{th}$ pattern of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Substitute $a_1=5$ and $d=2$: $$a_n = 5 + (n-1) \times 2 = 5 + 2n - 2 = 2n + 3$$ 9. **Find the number of dots in the 13$^{th}$ pattern:** $$a_{13} = 2 \times 13 + 3 = 26 + 3 = 29$$ **Final answers:** - a) The expression for the number of dots in the $n^{th}$ pattern is: $$a_n = 2n + 3$$ - b) The number of dots in the 13$^{th}$ pattern is: $$29$$