Geometric Sequences and Series

Step 1: Identify a₁ and r: a₁ = 3 r = 6/3 = 2

Step 2: Use the sum formula for n terms: Sₙ = a₁(1 - rⁿ)/(1 - r)

Step 3: Calculate S₅: S₅ = 3(1 - 2⁵)/(1 - 2) S₅ = 3(1 - 32)/(-1) S₅ = 3(-31)/(-1) S₅ = 93

Step 4: Verify by adding: 3 + 6 + 12 + 24 + 48 = 93 ✓

Answer: 93

Step 1: Verify it's geometric: r = 6/2 = 3 18/6 = 3 ✓

Step 2: Identify values: a₁ = 2, r = 3, n = 5

Step 3: Use sum formula: Sₙ = a₁(1 - rⁿ)/(1 - r) S₅ = 2(1 - 3⁵)/(1 - 3) S₅ = 2(1 - 243)/(-2) S₅ = 2(-242)/(-2) S₅ = 242

Step 4: Verify: 2 + 6 + 18 + 54 + 162 = 242 ✓

Answer: 242

Step 1: Use the sum formula: Sₙ = a₁(1 - rⁿ)/(1 - r)

Step 2: Substitute values: S₈ = 5(1 - (1/2)⁸)/(1 - 1/2) S₈ = 5(1 - 1/256)/(1/2)

Step 3: Simplify the numerator: 1 - 1/256 = 256/256 - 1/256 = 255/256

Step 4: Divide by 1/2: S₈ = 5 × (255/256) × 2 S₈ = 5 × 255/128 S₈ = 1275/128 S₈ ≈ 9.961

Answer: 1275/128 or approximately 9.961

Step 1: Identify a₁ and r: a₁ = 1 r = (1/3)/1 = 1/3

Step 2: Check if sum exists: For infinite series, sum exists if |r| < 1 |1/3| < 1 ✓

Step 3: Use infinite sum formula: S = a₁/(1 - r)

Step 4: Calculate: S = 1/(1 - 1/3) S = 1/(2/3) S = 3/2

Answer: 3/2 or 1.5

Step 1: Understand the motion:

• Falls 20 feet initially
• Bounces to 20(3/4) = 15 feet, then falls 15 feet
• Bounces to 15(3/4) = 11.25 feet, then falls 11.25 feet
• And so on...

Step 2: Set up the total distance: Distance = initial fall + 2(sum of all bounce heights) Distance = 20 + 2(15 + 11.25 + 8.4375 + ...)

Step 3: Identify the bounce series: First bounce: a₁ = 20(3/4) = 15 Common ratio: r = 3/4 This is an infinite geometric series

Step 4: Find sum of bounce heights: S = a₁/(1 - r) S = 15/(1 - 3/4) S = 15/(1/4) S = 60

Step 5: Calculate total distance: Total = 20 + 2(60) Total = 20 + 120 Total = 140 feet

Step 6: Alternative formula approach: Initial drop: 20 Up and down combined: 2 × 15/(1 - 3/4) = 2 × 60 = 120 Total: 140 feet ✓

Answer: 140 feet