1. **Determine if the sequence is arithmetic or geometric, find the ratio (r) or difference (d), and give the next 3 terms.** **a.** 28, 18, 8, ... - This is arithmetic because the difference between terms is constant. - Difference $d = 18 - 28 = -10$ - Next terms: $8 - 10 = -2$, $-2 - 10 = -12$, $-12 - 10 = -22$ **b.** -16, -6, 4, ... - Arithmetic sequence with difference $d = -6 - (-16) = 10$ - Next terms: $4 + 10 = 14$, $14 + 10 = 24$, $24 + 10 = 34$ **c.** -2, -10, -50, ... - Geometric sequence because ratio between terms is constant. - Ratio $r = \frac{-10}{-2} = 5$ - Next terms: $-50 \times 5 = -250$, $-250 \times 5 = -1250$, $-1250 \times 5 = -6250$ **d.** -3, -1, -\frac{1}{3}, ... - Geometric sequence. - Ratio $r = \frac{-1}{-3} = \frac{1}{3}$ - Next terms: $-\frac{1}{3} \times \frac{1}{3} = -\frac{1}{9}$, $-\frac{1}{9} \times \frac{1}{3} = -\frac{1}{27}$, $-\frac{1}{27} \times \frac{1}{3} = -\frac{1}{81}$ **e.** -1, 6, -36, ... - Geometric sequence. - Ratio $r = \frac{6}{-1} = -6$ - Next terms: $-36 \times -6 = 216$, $216 \times -6 = -1296$, $-1296 \times -6 = 7776$ **f.** 1, \frac{1}{2}, 0, ... - Arithmetic sequence. - Difference $d = \frac{1}{2} - 1 = -\frac{1}{2}$ - Next terms: $0 - \frac{1}{2} = -\frac{1}{2}$, $-\frac{1}{2} - \frac{1}{2} = -1$, $-1 - \frac{1}{2} = -\frac{3}{2}$ 2. **Find the common ratio $r$ algebraically for geometric sequences:** **a.** 3, -12, 48, ... - $r = \frac{-12}{3} = -4$ **b.** 0.01, 0.06, 0.36, ... - $r = \frac{0.06}{0.01} = 6$ **c.** $2x^8$, $8x^6$, $32x^4$, ... - $r = \frac{8x^6}{2x^8} = \frac{8}{2} \times \frac{x^6}{x^8} = 4x^{-2} = \frac{4}{x^2}$ **d.** 1, $x + 3$, $x^2 + 6x + 9$, ... - Note $x^2 + 6x + 9 = (x+3)^2$ - $r = \frac{x+3}{1} = x+3$ 3. **Find the indicated term in each sequence using geometric or arithmetic formulas:** **a.** 4, 12, 36, ... (geometric) - $r = \frac{12}{4} = 3$ - $n$th term formula: $a_n = a_1 r^{n-1}$ - $a_9 = 4 \times 3^{8} = 4 \times 6561 = 26244$ **b.** -2, 6, -18, ... (geometric) - $r = \frac{6}{-2} = -3$ - $a_9 = -2 \times (-3)^{8} = -2 \times 6561 = -13122$ **c.** -20, -10, -5, ... (arithmetic) - $d = -10 - (-20) = 10$ - $a_9 = a_1 + (n-1)d = -20 + 8 \times 5 = -20 + 40 = 20$ **d.** $\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{18}$, ... (geometric) - $r = \frac{1/6}{1/2} = \frac{1}{3}$ - $a_{12} = \frac{1}{2} \times \left(\frac{1}{3}\right)^{11} = \frac{1}{2} \times \frac{1}{3^{11}} = \frac{1}{2 \times 177147} = \frac{1}{354294}$ **e.** $3x^2$, $6x^4$, $12x^6$, ... (geometric) - $r = \frac{6x^4}{3x^2} = 2x^2$ - $a_7 = 3x^2 \times (2x^2)^{6} = 3x^2 \times 2^{6} x^{12} = 3 \times 64 \times x^{14} = 192 x^{14}$ **f.** 1, $2x$, $(2x)^2$, ... (geometric) - $r = 2x$ - $a_5 = 1 \times (2x)^{4} = (2x)^4 = 16 x^{4}$ 4. **The fifth term of a geometric sequence is 1875. First term is 3. Find the common ratio $r$.** - Formula: $a_n = a_1 r^{n-1}$ - $1875 = 3 r^{4}$ - Divide both sides by 3: $$\frac{1875}{3} = r^{4}$$ - $$625 = r^{4}$$ - Take fourth root: $$r = \pm 5$$ 5. **First term is 5, sixth term is 160. Find common ratio $r$.** - $a_6 = a_1 r^{5}$ - $160 = 5 r^{5}$ - Divide both sides by 5: $$\frac{160}{5} = r^{5}$$ - $$32 = r^{5}$$ - Take fifth root: $$r = 2$$