Infinite Geometric Series

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Infinite Geometric Series · Grade 12 · Worksheet 1

1. A wildlife biologist is studying the spread of a conservation message through a community. Each person who hears the message shares it with exactly 0.3 new people on average. If the initial outreach reached 800 people, what is the total number of people who will eventually hear this conservation message? Answer: 1142.8571428571429 Solution: Identify the first term (a) = 800 people initially reached Identify the common ratio (r) = 0.3 (each person shares with 0.3 new people) Use the infinite geometric series formula: S = a / (1 - r) Substitute the values: S = 800 / (1 - 0.3) Calculate denominator: 1 - 0.3 = 0.7 Divide: 800 / 0.7 =…
   Full step-by-step solution

   Step 1: Identify the first term (a) = 800 people initially reached Step 2: Identify the common ratio (r) = 0.3 (each person shares with 0.3 new people) Step 3: Use the infinite geometric series formula: S = a / (1 - r) Step 4: Substitute the values: S = 800 / (1 - 0.3) Step 5: Calculate denominator: 1 - 0.3 = 0.7 Step 6: Divide: 800 / 0.7 = 1142.8571428571429 Step 7: The total number of people who will eventually hear the message is approximately 1142.86 people.

2. Dr. Rodriguez is studying a new antidepressant medication that builds up in a patient's system over time. Each daily dose adds 80 mg to the bloodstream, but the body metabolizes 25% of the drug present each day. If a patient starts treatment and continues taking the medication indefinitely, what total drug concentration will approach in their system in the long run? Answer: 320 Solution: Identify the pattern - each new dose adds 80 mg, but only a fraction of previous doses remains After many doses, the total amount approaches: S = a / (1 - r) Where a = 80 mg (new dose) and r = 0.75 (fraction remaining after metabolism) Calculate: S = 80 / (1 - 0.75) = 80 / 0.25 = 320 The…
   Full step-by-step solution

   Step 1: Identify the pattern - each new dose adds 80 mg, but only a fraction of previous doses remains Step 2: After many doses, the total amount approaches: S = a / (1 - r) Step 3: Where a = 80 mg (new dose) and r = 0.75 (fraction remaining after metabolism) Step 4: Calculate: S = 80 / (1 - 0.75) = 80 / 0.25 = 320 Step 5: The long-term concentration approaches 320 mg The answer is 320.

3. ∑(n=1 to ∞) 9(0.8)^(n-1) = ? Answer: 45 Solution: The series is ∑(n=1 to ∞) 9(0.8)^(n-1) First term a = 9 Common ratio r = 0.8 Since |r| = |0.8| = 0.8 < 1, the series converges. Sum = a / (1 - r) Sum = 9 / (1 - 0.8) 1 - 0.8 = 0.2 Sum = 9 / 0.2 = 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 ∞) 9(0.8)^(n-1) First term a = 9 Common ratio 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 sum formula Sum = a / (1 - r) Sum = 9 / (1 - 0.8) Step 4: Calculate the denominator 1 - 0.8 = 0.2 Step 5: Divide to find the sum Sum = 9 / 0.2 = 45 The answer is 45.

4. A city is planning a new public transportation system where each new train line connects to existing lines, creating a geometric expansion pattern. The first line serves 12,000 daily commuters. Each subsequent line increases ridership by 25% of the previous line's ridership. If this expansion continues indefinitely, what is the maximum total daily ridership the system can support? Answer: 60000 Solution: Identify the first term: a = 12,000 Identify the common ratio: r = 0.25 Apply the infinite geometric series formula: S = a / (1 - r) Substitute the values: S = 12,000 / (1 - 0.25) Calculate denominator: 1 - 0.25 = 0.75 Divide: 12,000 / 0.75 = 60,000 The maximum total daily ridership is 60,000…
   Full step-by-step solution

   Step 1: Identify the first term: a = 12,000 Step 2: Identify the common ratio: r = 0.25 Step 3: Apply the infinite geometric series formula: S = a / (1 - r) Step 4: Substitute the values: S = 12,000 / (1 - 0.25) Step 5: Calculate denominator: 1 - 0.25 = 0.75 Step 6: Divide: 12,000 / 0.75 = 60,000 Step 7: The maximum total daily ridership is 60,000 commuters.

5. Olivia is monitoring the spread of a beneficial bacteria in a bioreactor used for waste treatment. Each hour, the bacteria break down 80% of the remaining waste in the reactor, leaving 20% of the waste untouched. If the initial waste load is 500 kg, and the bacteria continue this pattern indefinitely, what is the total amount of waste (in kg) that the bacteria will eventually break down? Answer: 500 Solution: Determine the first term (waste broken down in the first hour). The initial waste is 500 kg. The bacteria break down 80% of the waste each hour, so in the first hour, they break down 0.80 * 500 = 400 kg.
   Full step-by-step solution

   Step 1: Determine the first term (waste broken down in the first hour). The initial waste is 500 kg. The bacteria break down 80% of the waste each hour, so in the first hour, they break down 0.80 * 500 = 400 kg. Step 2: After the first hour, 500 - 400 = 100 kg remains. In the second hour, they break down 80% of the remaining 100 kg, which is 0.80 * 100 = 80 kg. Step 3: The waste broken down each hour forms a geometric sequence: 400, 80, 16, ... The common ratio r = 80 / 400 = 0.20 (since each hour they break down 20% of the previous hour's breakdown). Step 4: Since |r| = 0.20 < 1, the infinite geometric series converges. The sum S = a / (1 - r), where a = 400 and r = 0.20. Step 5: Calculate S = 400 / (1 - 0.20) = 400 / 0.80 = 500. Step 6: The total amount of waste the bacteria will eventually break down is 500 kg. The answer is 500.

6. Sophia is a materials scientist analyzing how a newly developed acoustic panel absorbs sound. In a controlled experiment, the panel absorbs 72 decibels of sound energy in the first second. In each subsequent second, it absorbs 20% of the sound energy it absorbed in the previous second. If this absorption pattern continues indefinitely, what is the total amount of sound energy, in decibels, that the panel will absorb over time? Answer: 90 Solution: Identify the series as an infinite geometric series. The first term a = 72 decibels (absorption in the first second). The common ratio r = 0.20 (each subsequent second absorbs 20% of the previous second's absorption).
   Full step-by-step solution

   Step 1: Identify the series as an infinite geometric series. The first term a = 72 decibels (absorption in the first second). The common ratio r = 0.20 (each subsequent second absorbs 20% of the previous second's absorption). Step 2: Check the condition for convergence. Since |r| = 0.20, which is less than 1, the series converges. Step 3: Apply the formula for the sum of an infinite geometric series: S = a / (1 - r). Step 4: Substitute the values: S = 72 / (1 - 0.20) = 72 / 0.80. Step 5: Simplify: 72 / 0.80 = 720 / 8 = 90. Step 6: The total sound energy absorbed over time is 90 decibels. The answer is 90.

7. ∑_{n=1}^∞ 3(0.6)^{n-1} = ? Answer: 7.5 Solution: sum from n= 1 to infinity of 3 * (0.6)^(n-1) a + a r + a r^2 + a r^3 + ... where the first term a = 3, and the common ratio r = 0.6.
   Full step-by-step solution

   We are given the infinite series: sum from n= 1 to infinity of 3 * (0.6)^(n-1) --- **Step 1: Identify the type of series** This is a geometric series of the form: a + a r + a r^2 + a r^3 + ... where the first term a = 3, and the common ratio r = 0.6. Check: For n = 1, term = 3 * (0.6)^(0) = 3 For n = 2, term = 3 * (0.6)^(1) = 1.8 For n = 3, term = 3 * (0.6)^(2) = 1.08 Indeed, each term is multiplied by 0.6 to get the next term. --- **Step 2: Condition for convergence** An infinite geometric series converges if |r| < 1. Here r = 0.6, which is less than 1, so the series converges. --- **Step 3: Formula for the sum of an infinite geometric series** The sum S of an infinite geometric series with first term a and common ratio r (|r| < 1) is: S = a / (1 - r) --- **Step 4: Substitute values** Here a = 3, r = 0.6 S = 3 / (1 - 0.6) S = 3 / 0.4 --- **Step 5: Simplify** 3 / 0.4 = 30 / 4 = 15 / 2 = 7.5 --- **Step 6: Conclusion** The sum of the infinite series is 7.5. --- **Final answer:** 7.5