Infinite Geometric Series

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Answer Key & Explanations

Infinite Geometric Series · Grade 12 · Worksheet 1

1. Emma is analyzing a fractal pattern formed by a series of nested squares. The largest square has a side length of 20 cm. Each subsequent square is drawn inside the previous one such that its vertices touch the midpoints of the sides of the larger square, creating a pattern that continues infinitely inward. Determine the total area of all the squares in this infinite geometric series. Answer: 800 Solution: The largest square has side length 20 cm, so its area is A1 = 20^2 = 400 sq cm. The next square's vertices touch the midpoints of the sides of the first square. This creates a square rotated 45 degrees inside.
   Full step-by-step solution

   Step 1: The largest square has side length 20 cm, so its area is A1 = 20^2 = 400 sq cm. Step 2: The next square's vertices touch the midpoints of the sides of the first square. This creates a square rotated 45 degrees inside. The side length of the inner square is the hypotenuse of a right triangle with legs equal to half the side of the outer square (10 cm each). So side length s2 = sqrt(10^2 + 10^2) = sqrt(200) = 10*sqrt(2) cm. Step 3: Area of the second square: A2 = (10*sqrt(2))^2 = 100 * 2 = 200 sq cm. Step 4: The pattern repeats: each new square's side length is 1/sqrt(2) times the previous square's side length. Therefore, each new area is (1/sqrt(2))^2 = 1/2 of the previous area. Step 5: This forms an infinite geometric series: 400 + 200 + 100 + 50 + ... Step 6: First term a = 400, common ratio r = 1/2. Since |r| = 1/2 < 1, the series converges. Step 7: Sum S = a / (1 - r) = 400 / (1 - 1/2) = 400 / (1/2) = 400 * 2 = 800. The total area of all squares is 800 sq cm.

2. ∑(n=1 to ∞) 14(0.8)^(n-1) = ? Answer: 70 Solution: Identify the first term (a) and common ratio (r) from the series. Here, a = 14 and r = 0.8. Check if the series converges.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) from the series. Here, a = 14 and r = 0.8. Step 2: Check if the series converges. An infinite geometric series converges if |r| < 1. Since |0.8| = 0.8 < 1, the series converges. Step 3: Apply the formula for the sum of an infinite convergent geometric series: S = a / (1 - r). Step 4: Substitute the values: S = 14 / (1 - 0.8) = 14 / 0.2. Step 5: Calculate the division: 14 ÷ 0.2 = 70. The answer is 70.

3. Tane is studying the population dynamics of an endangered tree fern species in a regenerating forest. He observes that each year, the number of new fronds produced by the population is exactly one-third of the number of fronds produced in the previous year. If the fern population produced 81 new fronds in the first year of his study, and this pattern of decline continues indefinitely, what is the total number of fronds that will ever be produced by this population from the first year onward? Answer: 121.5 Solution: Identify the series as an infinite geometric series. The first term (a) is the number of fronds produced in the first year: a = 81. The common ratio (r) is the factor by which the production decreases each year: r = 1/3.
   Full step-by-step solution

   Step 1: Identify the series as an infinite geometric series. The first term (a) is the number of fronds produced in the first year: a = 81. The common ratio (r) is the factor by which the production decreases each year: r = 1/3. Step 2: Check the condition for convergence. An infinite geometric series converges if |r| < 1. Here, |1/3| = 1/3 < 1, so the series converges. Step 3: Apply the formula for the sum of an infinite geometric series: S = a / (1 - r), where a is the first term and r is the common ratio. Step 4: Substitute the values: S = 81 / (1 - 1/3) Step 5: Simplify the denominator: 1 - 1/3 = 2/3 Step 6: Calculate the sum: S = 81 / (2/3) = 81 * (3/2) = 243/2 = 121.5 Step 7: The total number of fronds that will ever be produced is 121.5. The answer is 121.5.

4. A city is planning a new public transportation system where each new train line connects to existing lines. The first line serves 800,000 commuters daily. Each subsequent line serves 75% of the commuters of the previous line. If this expansion continues indefinitely, what is the total number of daily commuters the complete system will serve? Answer: 3200000 Solution: Identify the pattern - this is an infinite geometric series The first term is a = 800,000 The common ratio is r = 0.75 For an infinite geometric series, the sum formula is S = a / (1 - r) Substitute the values: S = 800,000 / (1 - 0.75) Calculate: 1 - 0.75 = 0.25 S = 800,000 / 0.25 S = 3,200,000…
   Full step-by-step solution

   Step 1: Identify the pattern - this is an infinite geometric series Step 2: The first term is a = 800,000 Step 3: The common ratio is r = 0.75 Step 4: For an infinite geometric series, the sum formula is S = a / (1 - r) Step 5: Substitute the values: S = 800,000 / (1 - 0.75) Step 6: Calculate: 1 - 0.75 = 0.25 Step 7: S = 800,000 / 0.25 Step 8: S = 3,200,000 The answer is 3,200,000.

5. ∑_{n=1}^{∞} 5(0.8)^{n-1} = ? Answer: 25 Solution: Identify the first term (a) and common ratio (r) from the series ∑_{n=1}^{∞} 5(0.8)^{n-1} The first term a = 5 Common ratio r = 0.8 Since |r| = |0.8| = 0.8 < 1, the series converges Sum = a/(1 - r) Sum = 5/(1 - 0.8) Sum = 5/0.2 Sum = 25 The answer is 25.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) from the series ∑_{n=1}^{∞} 5(0.8)^{n-1} The first term a = 5 Common ratio r = 0.8 Step 2: Check if the series converges Since |r| = |0.8| = 0.8 < 1, the series converges Step 3: Apply the infinite geometric series formula Sum = a/(1 - r) Sum = 5/(1 - 0.8) Step 4: Calculate the sum Sum = 5/0.2 Sum = 25 The answer is 25.

6. ∑(n=1 to ∞) 12(0.75)^(n-1) = ? Answer: 48 Solution: Identify the first term (a) and common ratio (r) from the series. Here, a = 12 and r = 0.75. Check if the series converges.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) from the series. Here, a = 12 and r = 0.75. Step 2: Check if the series converges. An infinite geometric series converges if |r| < 1. Since |0.75| = 0.75 < 1, the series converges. Step 3: Apply the formula for the sum of an infinite convergent geometric series: S = a / (1 - r). Step 4: Substitute the values: S = 12 / (1 - 0.75) = 12 / 0.25. Step 5: Calculate the division: 12 ÷ 0.25 = 48. The answer is 48.