1. Problem: For each sequence, find the next three terms and the rule for the terms. 2. Sequence a: $1, 2, 4, 8, 16, 32, \dots$ - Next three terms: $64, 128, 256$ - Rule: Each term is double the previous, $a_n = 2^{n-1}$ 3. Sequence b: $2, 10, 50, 250, \dots$ - Next three terms: $1250, 6250, 31250$ - Rule: Each term multiplied by 5, $b_n = 2 \times 5^{n-1}$ 4. Sequence c: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots$ - Next three terms: $\frac{1}{64}, \frac{1}{128}, \frac{1}{256}$ - Rule: Each term is half the previous, $c_n = \frac{1}{2^{n-1}}$ 5. Sequence d: $27, 9, 3, 1, \frac{1}{3}, \frac{1}{9}, \dots$ - Next three terms: $\frac{1}{27}, \frac{1}{81}, \frac{1}{243}$ - Rule: Each term is divided by 3, $d_n = \frac{27}{3^{n-1}}$ 6. Sequence e: $1, -2, 4, -8, 16, -32, \dots$ - Next three terms: $64, -128, 256$ - Rule: Alternates sign and doubles, $e_n = (-2)^{n-1}$ 7. Sequence f: $99, 9.9, 0.99, 0.099, \dots$ - Next three terms: $0.0099, 0.00099, 0.000099$ - Rule: Each term is divided by 10, $f_n = \frac{99}{10^{n-1}}$ 8. Second part sequences: a: $1, 4, 9, 16, 25, \dots$ - Next three: $36, 49, 64$ - Rule: Square numbers, $a_n = n^2$ b: $1, 8, 27, 64, 125, \dots$ - Next three: $216, 343, 512$ - Rule: Cube numbers, $b_n = n^3$ c: $2, 3, 5, 7, 11, 13, \dots$ - Next three: $17, 19, 23$ - Rule: Prime numbers d: $0, 1, 3, 6, 10, 15, \dots$ - Next three: $21, 28, 36$ - Rule: Triangular numbers, $d_n = \frac{n(n-1)}{2}$ e: $0, 1, 5, 14, 30, 55, \dots$ - Next three: $91, 140, 204$ - Rule: Figurate numbers: $e_n = \frac{n(n+1)(n+2)}{6}$ f: $\frac{1}{1}, \frac{2}{2}, \frac{3}{4}, \frac{4}{8}, \frac{5}{16}, \dots$ - Next three: $\frac{6}{32}, \frac{7}{64}, \frac{8}{128}$ - Rule: Numerator $n$, denominator $2^{n-2}$ for $n\ge2$; $f_n = \frac{n}{2^{n-2}}$ g: $1 \frac{1}{2}, 2 \frac{1}{3}, 3 \frac{1}{4}, 4 \frac{1}{5}, 5 \frac{1}{6}, \dots$ - Next three: $6 \frac{1}{7}, 7 \frac{1}{8}, 8 \frac{1}{9}$ - Rule: Mixed fraction $n \frac{1}{n+1}$ h: $0, 1, 1, 2, 3, 5, 8, \dots$ - Next three: $13, 21, 34$ - Rule: Fibonacci sequence, $h_n = h_{n-1} + h_{n-2}$ with $h_1=0, h_2=1$