1. **Stating the problem:** We are given two sequences and need to identify whether each is arithmetic or geometric, then write the explicit formula for the nth term $a_n$. 2. **Recall definitions:** - An **arithmetic sequence** has a constant difference $d$ between terms: $a_n = a_1 + (n-1)d$. - A **geometric sequence** has a constant ratio $r$ between terms: $a_n = a_1 \times r^{n-1}$. 3. **Sequence (a): 8, 32, 128, ...** - Check differences: $32 - 8 = 24$, $128 - 32 = 96$ (not constant). - Check ratios: $\frac{32}{8} = 4$, $\frac{128}{32} = 4$ (constant ratio). - So, sequence (a) is geometric with $a_1 = 8$ and $r = 4$. - Formula: $$a_n = 8 \times 4^{n-1}$$ 4. **Sequence (b): 2, 9, 16, ...** - Check differences: $9 - 2 = 7$, $16 - 9 = 7$ (constant difference). - Check ratios: $\frac{9}{2} = 4.5$, $\frac{16}{9} \approx 1.78$ (not constant). - So, sequence (b) is arithmetic with $a_1 = 2$ and $d = 7$. - Formula: $$a_n = 2 + (n-1) \times 7 = 7n - 5$$ **Final answers:** - (a) Geometric: $$a_n = 8 \times 4^{n-1}$$ - (b) Arithmetic: $$a_n = 7n - 5$$