Step 1: Find the common ratio: r = second term / first term r = 6/2 = 3

Step 2: Verify with third term: 18/6 = 3 ✓

Step 3: Find the next three terms: 4th term: 18 × 3 = 54 5th term: 54 × 3 = 162 6th term: 162 × 3 = 486

Answer: 54, 162, 486

Divide consecutive terms:

$r = \frac{15}{5} = 3$

Check: $\frac{45}{15} = 3$ ✓

Answer: Common ratio = $3$

Step 1: Identify the first term and common ratio: a₁ = 5 r = 15/5 = 3

Step 2: Use the formula for the nth term: aₙ = a₁ · rⁿ⁻¹

Step 3: Find the 8th term: a₈ = 5 · 3⁸⁻¹ a₈ = 5 · 3⁷ a₈ = 5 · 2187 a₈ = 10,935

Answer: 10,935

Step 1: Identify $a_1$ and $r$ $a_1 = 2, \quad r = \frac{6}{2} = 3$

Step 2: Use the explicit formula $a_n = a_1 \cdot r^{n-1}$

Step 3: Substitute $n = 8$ $a_8 = 2 \cdot 3^{8-1}$ $= 2 \cdot 3^7$ $= 2 \cdot 2187$ $= 4374$

Answer: $a_8 = 4374$

Step 1: Write formulas for both terms: a₃ = a₁ · r² = 12 a₆ = a₁ · r⁵ = 96

Step 2: Divide the second equation by the first: (a₁ · r⁵)/(a₁ · r²) = 96/12 r³ = 8

Step 3: Solve for r: r = ∛8 = 2

Step 4: Find a₁ using a₃ = 12: a₁ · r² = 12 a₁ · 2² = 12 a₁ · 4 = 12 a₁ = 3

Step 5: Verify with a₆: a₆ = 3 · 2⁵ = 3 · 32 = 96 ✓

Answer: a₁ = 3, r = 2

Step 1: Understand geometric mean: The geometric mean between a and b is √(ab) It forms a geometric sequence: a, geometric mean, b

Step 2: Calculate: Geometric mean = √(4 × 36) = √144 = 12

Step 3: Verify it forms a geometric sequence: Sequence: 4, 12, 36 Ratio: 12/4 = 3 36/12 = 3 ✓

Answer: 12

Given: $a_3 = 12$ and $a_6 = 96$

Step 1: Write equations using $a_n = a_1 \cdot r^{n-1}$ $a_3: \quad 12 = a_1 \cdot r^2$ $a_6: \quad 96 = a_1 \cdot r^5$

Step 2: Divide the second by the first $\frac{96}{12} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2}$ $8 = r^3$ $r = 2$

Step 3: Find $a_1$ using $a_3 = 12$ $12 = a_1 \cdot 2^2$ $12 = 4a_1$ $a_1 = 3$

Check: Sequence is $3, 6, 12, 24, 48, 96, ...$ ✓

Answer: $a_1 = 3$, $r = 2$

Step 1: Identify the geometric sequence: Initial population: a₁ = 100 Common ratio: r = 2 (doubles)

Step 2: Write the explicit formula: aₙ = a₁ · rⁿ⁻¹ aₙ = 100 · 2ⁿ⁻¹

Step 3: Find population after 10 hours (11th term, since n=1 is initial): a₁₁ = 100 · 2¹¹⁻¹ a₁₁ = 100 · 2¹⁰ a₁₁ = 100 · 1024 a₁₁ = 102,400

Step 4: Alternative interpretation (after 10 hours, so 10 doublings): If we consider the initial as time 0: After 10 hours: 100 · 2¹⁰ = 102,400

Step 5: Verify doubling pattern: Hour 0: 100 Hour 1: 200 (×2) Hour 2: 400 (×2) ... Hour 10: 102,400

Answer: Formula: aₙ = 100 · 2ⁿ⁻¹ After 10 hours: 102,400 bacteria