1. **Identify which sequences are geometric and find the common ratio.** A geometric sequence has a constant ratio $r$ between consecutive terms: $$r = \frac{t_{n+1}}{t_n}$$ **a)** Sequence: 4, 12, 36, 108, ... Calculate ratios: $$\frac{12}{4} = 3, \quad \frac{36}{12} = 3, \quad \frac{108}{36} = 3$$ All ratios equal 3, so it is geometric with $r=3$. **b)** Sequence: 135, -45, 15, -5, ... Calculate ratios: $$\frac{-45}{135} = -\frac{1}{3}, \quad \frac{15}{-45} = -\frac{1}{3}, \quad \frac{-5}{15} = -\frac{1}{3}$$ All ratios equal $-\frac{1}{3}$, so it is geometric with $r=-\frac{1}{3}$. **c)** Sequence: 3, 6, 9, 12, ... Calculate ratios: $$\frac{6}{3} = 2, \quad \frac{9}{6} = 1.5, \quad \frac{12}{9} = \frac{4}{3}$$ Ratios are not constant, so it is not geometric. **d)** Sequence: $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, ...$ Calculate ratios: $$\frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}, \quad \frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{2}, \quad \frac{\frac{1}{32}}{\frac{1}{16}} = \frac{1}{2}$$ All ratios equal $\frac{1}{2}$, so it is geometric with $r=\frac{1}{2}$. **Final answers:** - a) Geometric, $r=3$ - b) Geometric, $r=-\frac{1}{3}$ - c) Not geometric - d) Geometric, $r=\frac{1}{2}$