1. **State the problem:** We need to classify each given sequence as either an arithmetic sequence or a geometric sequence. 2. **Recall definitions:** - An **arithmetic sequence** has a constant difference between consecutive terms: $a_{n+1} - a_n = d$ (constant). - A **geometric sequence** has a constant ratio between consecutive terms: $\frac{a_{n+1}}{a_n} = r$ (constant). 3. **Analyze each sequence:** - Sequence (90, 87, 84, 81, 78, ...): Calculate differences: $87 - 90 = -3$, $84 - 87 = -3$, $81 - 84 = -3$. Differences are constant $-3$, so this is **arithmetic**. - Sequence (4, 12, 36, 108, 324, ...): Calculate ratios: $\frac{12}{4} = 3$, $\frac{36}{12} = 3$, $\frac{108}{36} = 3$. Ratios are constant $3$, so this is **geometric**. - Sequence (8, 1, \frac{1}{8}, \frac{1}{64}, \frac{1}{512}, ...): Calculate ratios: $\frac{1}{8} = \frac{1}{8}$, $\frac{\frac{1}{8}}{1} = \frac{1}{8}$, $\frac{\frac{1}{64}}{\frac{1}{8}} = \frac{1}{8}$. Ratios are constant $\frac{1}{8}$, so this is **geometric**. - Sequence (3, 10, 17, 24, 31, ...): Calculate differences: $10 - 3 = 7$, $17 - 10 = 7$, $24 - 17 = 7$. Differences are constant $7$, so this is **arithmetic**. - Sequence (1, 1.4, 1.8, 2.2, 2.6, ...): Calculate differences: $1.4 - 1 = 0.4$, $1.8 - 1.4 = 0.4$, $2.2 - 1.8 = 0.4$. Differences are constant $0.4$, so this is **arithmetic**. - Sequence (3, 1.5, 0.75, 0.375, ...): Calculate ratios: $\frac{1.5}{3} = 0.5$, $\frac{0.75}{1.5} = 0.5$, $\frac{0.375}{0.75} = 0.5$. Ratios are constant $0.5$, so this is **geometric**. 4. **Final classification:** - Arithmetic sequences: (90, 87, 84, 81, 78, ...), (3, 10, 17, 24, 31, ...), (1, 1.4, 1.8, 2.2, 2.6, ...) - Geometric sequences: (4, 12, 36, 108, 324, ...), (8, 1, \frac{1}{8}, \frac{1}{64}, \frac{1}{512}, ...), (3, 1.5, 0.75, 0.375, ...) This completes the sorting of sequences into arithmetic and geometric categories.