Infinite Geometric Series

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Answer Key & Explanations

Infinite Geometric Series · Grade 12 · Worksheet 2

1. A wildlife biologist is studying the spread of a rare bird species through a forest ecosystem. Each breeding pair successfully establishes a new nesting territory with a probability of 0.3, and these new territories then become sources for further expansion. If the initial population consists of 80 breeding pairs, what is the theoretical maximum total number of breeding pairs that could eventually be supported in this forest under ideal conditions? Answer: 400 Solution: Identify the pattern as an infinite geometric series - Initial breeding pairs: 80 - Each pair establishes new territories with probability 0.3 - This means each generation adds 0.3 times the current number Total = 80 + 80(0.3) + 80(0.3)^2 + 80(0.3)^3 + ...
   Full step-by-step solution

   Step 1: Identify the pattern as an infinite geometric series - Initial breeding pairs: 80 - Each pair establishes new territories with probability 0.3 - This means each generation adds 0.3 times the current number Step 2: Write the series Total = 80 + 80(0.3) + 80(0.3)^2 + 80(0.3)^3 + ... Step 3: Apply the infinite geometric series formula Sum = a / (1 - r), where a = first term, r = common ratio First term a = 80 Common ratio r = 0.3 Step 4: Calculate the sum Sum = 80 / (1 - 0.3) = 80 / 0.7 = 800 / 7 = 400 Step 5: Interpret the result The theoretical maximum total number of breeding pairs is 400. The answer is 400.

2. 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.

3. Emma is investigating the long-term accumulation of a new eco-friendly cleaning agent in a lake. Each month, a factory releases 500 kilograms of the agent into the lake. The lake's natural filtration system removes 40% of the agent present at the beginning of each month. If the factory continues this monthly release indefinitely, what total mass of the cleaning agent will accumulate in the lake in the long run? Answer: 1250 Solution: Identify the pattern. Each month, 500 kg is added, but 40% is removed, meaning 60% remains from the previous month. The amount present just after the nth release forms a geometric series.
   Full step-by-step solution

   Step 1: Identify the pattern. Each month, 500 kg is added, but 40% is removed, meaning 60% remains from the previous month. The amount present just after the nth release forms a geometric series. Step 2: Write the series for the total mass after infinite months. The first month: 500 kg. Second month: 500 kg (new) + 500(0.6) kg (remaining from first month). Third month: 500 + 500(0.6) + 500(0.6)^2, and so on. Step 3: This is an infinite geometric series: S = 500 + 500(0.6) + 500(0.6)^2 + 500(0.6)^3 + ... Step 4: Check for convergence. The common ratio r = 0.6. Since |0.6| < 1, the series converges. Step 5: Use the formula for the sum of an infinite geometric series: S = a / (1 - r), where a = 500 and r = 0.6. Step 6: Calculate: S = 500 / (1 - 0.6) = 500 / 0.4 = 1250. The total mass of cleaning agent that will accumulate in the lake in the long run is 1250 kilograms.

4. ∑(n=1 to ∞) 15(0.9)^(n-1) = ? Answer: 150 Solution: The series is ∑(n=1 to ∞) 15(0.9)^(n-1) First term a = 15 Common ratio r = 0.9 Since |r| = |0.9| = 0.9 < 1, the series converges.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) The series is ∑(n=1 to ∞) 15(0.9)^(n-1) First term a = 15 Common ratio r = 0.9 Step 2: Check convergence condition Since |r| = |0.9| = 0.9 < 1, the series converges. Step 3: Apply the infinite geometric series sum formula Sum = a / (1 - r) Sum = 15 / (1 - 0.9) Step 4: Calculate the denominator 1 - 0.9 = 0.1 Step 5: Divide to find the sum Sum = 15 / 0.1 = 150 The answer is 150.

5. ∑(n=1 to ∞) 5(0.8)^(n-1) = ? Answer: 25 Solution: The series is 5(0.8)^(n-1), so a = 5 and r = 0.8 Since |r| = |0.8| = 0.8 < 1, the series converges Sum = a / (1 - r) Sum = 5 / (1 - 0.8) 1 - 0.8 = 0.2 5 / 0.2 = 25 The answer is 25.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) The series is 5(0.8)^(n-1), so a = 5 and r = 0.8 Step 2: Check convergence condition 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 denominator 1 - 0.8 = 0.2 Step 5: Divide to find the sum 5 / 0.2 = 25 The answer is 25.

6. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration follows the pattern: 120 mg/L, then 60 mg/L, then 30 mg/L, and continues decreasing in this manner indefinitely. If this pattern represents an infinite geometric series, what is the total drug concentration that will eventually be present in the bloodstream? Answer: 240 Solution: We are told the drug concentration follows the pattern: 120 mg/L, 60 mg/L, 30 mg/L, and so on, decreasing indefinitely in the same manner. Identify the type of series.
   Full step-by-step solution

   We are told the drug concentration follows the pattern: 120 mg/L, 60 mg/L, 30 mg/L, and so on, decreasing indefinitely in the same manner. Step 1: Identify the type of series. The problem says this is an infinite geometric series. A geometric series has a constant ratio between consecutive terms. Step 2: Find the common ratio. Divide the second term by the first term: 60 / 120 = 1/2 Check the third term divided by the second term: 30 / 60 = 1/2 So the common ratio r = 1/2. Step 3: Recall the sum formula for an infinite geometric series. The sum S of an infinite geometric series with first term a and common ratio r (where |r| < 1) is: S = a / (1 - r) Step 4: Apply the formula. Here, a = 120, r = 1/2. S = 120 / (1 - 1/2) S = 120 / (1/2) S = 120 * (2/1) S = 240 Step 5: Interpret the result. The total drug concentration that will eventually be present in the bloodstream is 240 mg/L. This means if you add up 120 + 60 + 30 + 15 + ... forever, the sum approaches 240. Final answer: 240

7. ∑_{n=1}^{∞} 7(0.3)^{n-1} = ? Answer: 10 Solution: Identify the first term (a) and common ratio (r) from the series. Here, a = 7 and r = 0.3. Check the condition for convergence: |r| < 1.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) from the series. Here, a = 7 and r = 0.3. Step 2: Check the condition for convergence: |r| < 1. |0.3| = 0.3 < 1, so the series converges. Step 3: Apply the formula for the sum of an infinite geometric series: S = a / (1 - r). S = 7 / (1 - 0.3) Step 4: Calculate the denominator: 1 - 0.3 = 0.7. Step 5: Divide: 7 / 0.7 = 10. The sum of the infinite geometric series is 10.