Geometric Sequences E2Eaec

1. **State the problem:** Determine which of the given sequences are geometric sequences. 2. **Recall the definition:** A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant ratio $r$. 3. **Check each sequence:** - Sequence 1: $1, 4, 16, 64, \ldots$ Calculate ratio: $\frac{4}{1} = 4$, $\frac{16}{4} = 4$, $\frac{64}{16} = 4$ All ratios equal $4$, so this is geometric. - Sequence 2: $-2, -8, -32, -128, \ldots$ Calculate ratio: $\frac{-8}{-2} = 4$, $\frac{-32}{-8} = 4$, $\frac{-128}{-32} = 4$ All ratios equal $4$, so this is geometric. - Sequence 3: $48, 24, 12, 6, \ldots$ Calculate ratio: $\frac{24}{48} = \frac{1}{2}$, $\frac{12}{24} = \frac{1}{2}$, $\frac{6}{12} = \frac{1}{2}$ All ratios equal $\frac{1}{2}$, so this is geometric. - Sequence 4: $1, 2, 3, 5, \ldots$ Calculate ratio: $\frac{2}{1} = 2$, $\frac{3}{2} = 1.5$, $\frac{5}{3} \approx 1.67$ Ratios are not equal, so not geometric. - Sequence 5: $2, 6, 10, 14, \ldots$ Calculate ratio: $\frac{6}{2} = 3$, $\frac{10}{6} \approx 1.67$, $\frac{14}{10} = 1.4$ Ratios are not equal, so not geometric. - Sequence 6: $1, 0.2, 0.04, 0.008, \ldots$ Calculate ratio: $\frac{0.2}{1} = 0.2$, $\frac{0.04}{0.2} = 0.2$, $\frac{0.008}{0.04} = 0.2$ All ratios equal $0.2$, so this is geometric. - Sequence 7: $7, 70, 700, 7000, \ldots$ Calculate ratio: $\frac{70}{7} = 10$, $\frac{700}{70} = 10$, $\frac{7000}{700} = 10$ All ratios equal $10$, so this is geometric. - Sequence 8: $3, 9, 27, 81, \ldots$ Calculate ratio: $\frac{9}{3} = 3$, $\frac{27}{9} = 3$, $\frac{81}{27} = 3$ All ratios equal $3$, so this is geometric. - Sequence 9: $2, 22, 222, 2222, \ldots$ Calculate ratio: $\frac{22}{2} = 11$, $\frac{222}{22} = 10.09$, $\frac{2222}{222} = 10.01$ Ratios are not equal, so not geometric. - Sequence 10: $3, 25, 729, 534361, \ldots$ Calculate ratio: $\frac{25}{3} \approx 8.33$, $\frac{729}{25} = 29.16$, $\frac{534361}{729} \approx 732.9$ Ratios are not equal, so not geometric. **Final answer:** The geometric sequences are: - $1, 4, 16, 64, \ldots$ - $-2, -8, -32, -128, \ldots$ - $48, 24, 12, 6, \ldots$ - $1, 0.2, 0.04, 0.008, \ldots$ - $7, 70, 700, 7000, \ldots$ - $3, 9, 27, 81, \ldots$