1. **Stating the problem:** We are given two sequences and need to write recursive formulas for each. 2. **Sequence 1:** 6, 2, 2/3, 2/9, ... with initial term $a_1 = 6$. 3. **Sequence 2:** 43, 30, 17, 4, ... with initial term $a_1 = 43$. 4. **Recall recursive formulas:** - For an arithmetic sequence, the recursive formula is $a_n = a_{n-1} + d$ where $d$ is the common difference. - For a geometric sequence, the recursive formula is $a_n = r \times a_{n-1}$ where $r$ is the common ratio. 5. **Analyze Sequence 1:** - Calculate ratio between terms: $\frac{2}{6} = \frac{1}{3}$, $\frac{2/3}{2} = \frac{1}{3}$, $\frac{2/9}{2/3} = \frac{1}{3}$. - Since the ratio is constant $r = \frac{1}{3}$, Sequence 1 is geometric. - Recursive formula: $$a_n = \frac{1}{3} a_{n-1} \text{ for } n \geq 2$$ 6. **Analyze Sequence 2:** - Calculate differences: $30 - 43 = -13$, $17 - 30 = -13$, $4 - 17 = -13$. - Since the difference is constant $d = -13$, Sequence 2 is arithmetic. - Recursive formula: $$a_n = a_{n-1} - 13 \text{ for } n \geq 2$$ **Final answers:** - Sequence 1: $a_1 = 6$, $a_n = \frac{1}{3} a_{n-1}$ for $n \geq 2$. - Sequence 2: $a_1 = 43$, $a_n = a_{n-1} - 13$ for $n \geq 2$. These recursive formulas allow you to find any term in the sequences by using the previous term.