Infinite Geometric Series

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Infinite Geometric Series · Grade 12 · Worksheet 2

1. A pharmaceutical company is modeling the decay of a radioactive tracer in the human body. The amount of tracer remaining after each 24-hour period follows a geometric sequence where each term is 0.75 times the previous term. If the initial dose is 80 milligrams, what is the total amount of tracer that will eventually be present in the body over infinite time? Answer: 320 Solution: Identify the first term (a) of the geometric series. The initial dose is 80 mg, so a = 80. Identify the common ratio (r).
   Full step-by-step solution

   Step 1: Identify the first term (a) of the geometric series. The initial dose is 80 mg, so a = 80. Step 2: Identify the common ratio (r). Each term is 0.75 times the previous term, so r = 0.75. Step 3: Check if the series converges. Since |r| = 0.75 < 1, the infinite geometric series converges. Step 4: Apply the formula for the sum of an infinite geometric series: S = a/(1 - r). Step 5: Substitute the values: S = 80/(1 - 0.75) = 80/0.25. Step 6: Calculate: 80/0.25 = 320. The total amount of tracer that will eventually be present in the body is 320 milligrams.

2. Mere is studying a fractal pattern formed by a series of nested circles. The largest circle has a radius of 12 cm. Each subsequent circle is drawn inside the previous one such that its diameter is exactly half the diameter of the larger circle, and this pattern continues infinitely inward. Determine the total area of all the circles in this infinite geometric series, giving your answer in terms of π. Answer: 192π Solution: The largest circle has radius r1 = 12 cm. Its area is A1 = π * (12)^2 = 144π sq cm. The next circle has a diameter half of the previous, so its radius is half of 12, which is r2 = 6 cm.
   Full step-by-step solution

   Step 1: The largest circle has radius r1 = 12 cm. Its area is A1 = π * (12)^2 = 144π sq cm. Step 2: The next circle has a diameter half of the previous, so its radius is half of 12, which is r2 = 6 cm. Its area is A2 = π * (6)^2 = 36π sq cm. Step 3: The third circle has radius r3 = 3 cm. Its area is A3 = π * (3)^2 = 9π sq cm. Step 4: This forms an infinite geometric series: 144π + 36π + 9π + 2.25π + ... Step 5: First term a = 144π. Common ratio r = A2/A1 = 36π / 144π = 1/4. Step 6: Since |r| = 1/4 < 1, the series converges. Step 7: Sum S = a / (1 - r) = 144π / (1 - 1/4) = 144π / (3/4) = 144π * (4/3) = 192π. The total area of all circles is 192π sq cm.

3. Noah is a theoretical physicist modeling the behavior of a quantum harmonic oscillator. He discovers that the energy levels of the oscillator follow a specific pattern: the ground state energy is 48 femtojoules (fJ), and each successive excited state has an energy equal to 60% of the previous state's energy. If this infinite sequence of energy levels continues, what is the total energy (in fJ) of all possible states combined? Answer: 120 Solution: Identify the first term and common ratio. The ground state energy is 48 fJ, so a = 48. Each successive state has 60% of the previous state's energy, so r = 0.6.
   Full step-by-step solution

   Step 1: Identify the first term and common ratio. The ground state energy is 48 fJ, so a = 48. Each successive state has 60% of the previous state's energy, so r = 0.6. Step 2: Check convergence. |r| = |0.6| = 0.6, which is less than 1. Therefore, the infinite geometric series converges. Step 3: Apply the infinite geometric series sum formula. S = a / (1 - r) = 48 / (1 - 0.6) = 48 / 0.4 = 120. Step 4: State the final answer. The total energy of all possible states combined is 120 fJ.

4. ∑(n=1 to ∞) 2(0.8)^(n-1) = ? Answer: 10 Solution: Identify the first term (a) and common ratio (r) from the series ∑(n=1 to ∞) 2(0.8)^(n-1) First term a = 2 Common ratio r = 0.8 Since |r| = |0.8| = 0.8 < 1, the series converges Sum = a / (1 - r) Sum = 2 / (1 - 0.8) 1 - 0.8 = 0.2 Sum = 2 / 0.2 = 10 The answer is 10.
   Full step-by-step solution

   Step 1: Identify the first term (a) and common ratio (r) from the series ∑(n=1 to ∞) 2(0.8)^(n-1) First term a = 2 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 sum formula Sum = a / (1 - r) Sum = 2 / (1 - 0.8) Step 4: Calculate the denominator 1 - 0.8 = 0.2 Step 5: Divide to find the sum Sum = 2 / 0.2 = 10 The answer is 10.

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 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.