Infinite Geometric Series

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Infinite Geometric Series · Grade 12 · Worksheet 3

1. A city is planning a new public transportation system where each new train line connects to existing lines, creating a geometric growth pattern. The first line serves 12,000 daily commuters. Each subsequent line serves 75% of the previous line's ridership. If the city builds lines indefinitely following this pattern, what is the total number of daily commuters the entire system will eventually serve? Answer: 48000 Solution: Identify the first term: a = 12,000 Identify the common ratio: r = 0.75 Check if the series converges: |r| = 0.75 < 1, so it converges Apply the infinite geometric series formula: S = a / (1 - r) Substitute values: S = 12,000 / (1 - 0.75) Calculate denominator: 1 - 0.75 = 0.25 Divide: 12,000 /…
   Full step-by-step solution

   Step 1: Identify the first term: a = 12,000 Step 2: Identify the common ratio: r = 0.75 Step 3: Check if the series converges: |r| = 0.75 < 1, so it converges Step 4: Apply the infinite geometric series formula: S = a / (1 - r) Step 5: Substitute values: S = 12,000 / (1 - 0.75) Step 6: Calculate denominator: 1 - 0.75 = 0.25 Step 7: Divide: 12,000 / 0.25 = 48,000 Step 8: The total number of daily commuters is 48,000.

2. ∑(n=1 to ∞) 18(0.6)^(n-1) = ? Answer: 45 Solution: The series is ∑(n=1 to ∞) 18(0.6)^(n-1) First term a = 18 Common ratio r = 0.6 Since |r| = |0.6| = 0.6 < 1, the series converges. Sum = a / (1 - r) Sum = 18 / (1 - 0.6) 1 - 0.6 = 0.4 Sum = 18 / 0.4 = 45 The answer is 45.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) The series is ∑(n=1 to ∞) 18(0.6)^(n-1) First term a = 18 Common ratio r = 0.6 Step 2: Check convergence condition Since |r| = |0.6| = 0.6 < 1, the series converges. Step 3: Apply the infinite geometric series sum formula Sum = a / (1 - r) Sum = 18 / (1 - 0.6) Step 4: Calculate the denominator 1 - 0.6 = 0.4 Step 5: Divide to find the sum Sum = 18 / 0.4 = 45 The answer is 45.

3. ∑(n=1 to ∞) 17(0.7)^(n-1) = ? Answer: 56.67 Solution: Identify the first term (a) and common ratio (r) from the series. Here, a = 17 and r = 0.7. 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 = 17 and r = 0.7. Step 2: Check if the series converges. An infinite geometric series converges if |r| < 1. Since |0.7| = 0.7 < 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 = 17 / (1 - 0.7) = 17 / 0.3. Step 5: Calculate the division: 17 ÷ 0.3 = 56.666..., which rounds to 56.67. The answer is 56.67.

4. Olivia is analyzing the long-term concentration of a new anti-inflammatory drug in a patient's bloodstream. The patient receives an initial intravenous injection of 75 mg of the drug. At the end of each 6-hour interval, exactly 35% of the drug that was present at the start of that interval remains in the bloodstream, and the patient receives a new maintenance dose of 75 mg. If this dosing pattern continues indefinitely, what total amount of drug (in mg) will be present in the patient's system in the long run? Answer: 115.38 Solution: Identify the pattern. Let A_n be the amount of drug in the bloodstream immediately after the nth dose. A_1 = 75 mg (the initial dose).
   Full step-by-step solution

   Step 1: Identify the pattern. Let A_n be the amount of drug in the bloodstream immediately after the nth dose. A_1 = 75 mg (the initial dose). After 6 hours, 35% of A_1 remains, so amount = 0.35 * 75. Then a new dose of 75 mg is added. So A_2 = 0.35 * 75 + 75. Similarly, A_3 = 0.35 * (0.35 * 75 + 75) + 75 = 75 + 75*0.35 + 75*(0.35)^2. In general, after many doses, the amount just after a dose is: A_n = 75 + 75*0.35 + 75*(0.35)^2 + ... + 75*(0.35)^(n-1). Step 2: This is a finite geometric series for n doses, but as n approaches infinity, it becomes an infinite geometric series: Total amount = 75 + 75(0.35) + 75(0.35)^2 + 75(0.35)^3 + ... Step 3: Check convergence. The common ratio r = 0.35. Since |0.35| < 1, the series converges. Step 4: Apply the infinite geometric series sum formula. First term a = 75. Common ratio r = 0.35. Sum S = a / (1 - r) = 75 / (1 - 0.35) = 75 / 0.65. Step 5: Calculate. 75 / 0.65 = 75 / (65/100) = 75 * (100/65) = 7500/65 = 1500/13 ≈ 115.3846. Step 6: Round to two decimal places (since drug amounts are often given this way): 115.38 mg. The total amount of drug that will be present in the patient's system in the long run is approximately 115.38 mg.

5. A pharmaceutical company is testing a new drug that has a half-life of 8 hours in the human body. If a patient takes a 200 mg dose, and the drug's concentration decreases geometrically, what is the total amount of the drug that will ever be present in the patient's system over infinite time? Answer: 400 Solution: The half-life is 8 hours. Initial dose = 200 mg. The drug concentration decreases geometrically — meaning after each half-life, it's multiplied by 1/2.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The half-life is 8 hours. Initial dose = 200 mg. The drug concentration decreases geometrically — meaning after each half-life, it's multiplied by 1/2. We want the **total** amount ever present in the system over infinite time, assuming the patient only takes one dose. --- **Step 2: Model the drug amount over time** At time 0: amount = 200 mg After 8 hours: amount = 200 × (1/2) = 100 mg After 16 hours: amount = 200 × (1/2)^2 = 50 mg After 24 hours: amount = 200 × (1/2)^3 = 25 mg And so on. --- **Step 3: Write the total amount as an infinite series** Total amount over infinite time = 200 + 200 × (1/2) + 200 × (1/2)^2 + 200 × (1/2)^3 + ... Factor out 200: Total = 200 × [ 1 + 1/2 + (1/2)^2 + (1/2)^3 + ... ] --- **Step 4: Recognize the geometric series** The series inside the brackets is: 1 + r + r^2 + r^3 + ... where r = 1/2. Sum of infinite geometric series = 1 / (1 - r) when |r| < 1. So: 1 + 1/2 + (1/2)^2 + ... = 1 / (1 - 1/2) = 1 / (1/2) = 2. --- **Step 5: Multiply back** Total amount = 200 × 2 = 400 mg. --- **Step 6: Interpret the result** The total drug exposure over infinite time from a single 200 mg dose is 400 mg·hours if thinking in terms of area under the curve, but here it's the "sum of all instantaneous amounts" in a discrete sense — actually, in pharmacokinetics, this is the total drug exposure if you could sum all the mass present at each instant, but the problem likely means: the drug decays geometrically in mass per half-life, and we sum over all time — which gives 400 mg total cumulative presence. --- **Final Answer:** 400

6. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration follows the pattern C(t) = 8 + 4e^(-0.2t) mg/L, where t is time in hours. The company wants to determine the long-term steady-state concentration that patients will approach. What is the limit of C(t) as t approaches infinity? Answer: 8 Solution: C(t) = 8 + 4 * e^(-0.2 * t) Identify the exponential term. The term 4 * e^(-0.2 * t) contains the exponential function with a negative exponent: -0.2 * t. Analyze the behavior of e^(-0.2 * t) as t → ∞.
   Full step-by-step solution

   Let's find the limit of C(t) as t approaches infinity. We have: C(t) = 8 + 4 * e^(-0.2 * t) Step 1: Identify the exponential term. The term 4 * e^(-0.2 * t) contains the exponential function with a negative exponent: -0.2 * t. Step 2: Analyze the behavior of e^(-0.2 * t) as t → ∞. Since -0.2 is negative, as t gets very large, the exponent -0.2 * t becomes a large negative number. We know that e raised to a large negative number approaches 0. So, e^(-0.2 * t) → 0 as t → ∞. Step 3: Substitute the limit of the exponential term into C(t). As t → ∞: C(t) = 8 + 4 * e^(-0.2 * t) → 8 + 4 * 0 → 8 + 0 → 8 Step 4: Conclusion. The long-term steady-state concentration is 8 mg/L. Answer: 8

7. A pharmaceutical company is testing a new drug that has a half-life of 8 hours in the human body. If a patient takes a 200 mg dose, and the drug's concentration decreases geometrically over time, what is the total amount of the drug that will eventually be present in the patient's system from this single dose? Answer: 400 mg Solution: When a quantity decreases by a constant ratio over equal time periods, we can model this using an infinite geometric series. The sum of such a series exists when the common ratio has an absolute value less than 1, and can be calculated using the formula for the sum of an infinite geometric series.
   Full step-by-step solution

   When a quantity decreases by a constant ratio over equal time periods, we can model this using an infinite geometric series. The sum of such a series exists when the common ratio has an absolute value less than 1, and can be calculated using the formula for the sum of an infinite geometric series. This concept applies to various real-world scenarios like drug elimination, radioactive decay, and depreciation of assets.