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Geometric Sequences

Grade 12 · Geometry · Worksheet 1

  1. Emma is a microbiologist studying the growth of a bacterial colony in a laboratory. She observes that the colony follows a geometric growth pattern. Initially, there are 7 bacterial cells. Every 4 hours, the number of cells triples. Emma wants to determine the number of bacterial cells present after 24 hours (the 7th term of the sequence) and the total number of cells that have existed over the entire 24-hour period (sum of the first 7 terms). Find the number of cells at 24 hours and the total number of cells over the first 7 time intervals. Answer: ______________
  2. Aroha is an environmental scientist monitoring the population of a rare bird species in a protected wetland. She observes that the population follows a geometric growth pattern due to successful conservation efforts. In the first year of her study, there are 9 birds. Each subsequent year, the population increases by a factor of 4 times the previous year's population. Aroha wants to predict the bird population after 10 years (the 10th term of the sequence) and the total number of birds that have lived in the wetland over the first 10 years. Find the population in the 10th year and the total population over the first 10 years. Answer: ______________
  3. Emma is a botanist studying the population growth of a rare fern species in a protected reserve. She notices that the number of ferns in a specific plot follows a geometric pattern. In the first year, there are 7 ferns. Each subsequent year, the number of ferns triples compared to the previous year. Emma needs to report the number of ferns after 9 years (the 9th term of the sequence) and the total number of ferns that have appeared over the first 9 years. Find the number of ferns in the 9th year and the total number of ferns over the first 9 years. Answer: ______________
  4. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration C(t) in mg/L follows the function C(t) = 80e^(-0.15t) + 20e^(-0.03t), where t is time in hours. The therapeutic window for this drug is between 40 mg/L and 85 mg/L. Determine the total time interval during which the drug concentration remains within the therapeutic window. Answer: ______________
  5. Matiu is a marine biologist studying the population growth of a rare species of coral on a protected reef. He notes that the coral covers an area that grows geometrically each year. In the first year, the coral covers 24 square metres. Each subsequent year, the area covered increases by a factor of 2.5 times the previous year's area. Matiu needs to report the area covered after 10 years (the 10th term of the sequence) and the total area covered over the first 10 years for a conservation grant application. Find the area covered in the 10th year and the total area covered over the first 10 years. Answer: ______________
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Answer Key & Explanations

Geometric Sequences · Grade 12 · Worksheet 1

  1. Emma is a microbiologist studying the growth of a bacterial colony in a laboratory. She observes that the colony follows a geometric growth pattern. Initially, there are 7 bacterial cells. Every 4 hours, the number of cells triples. Emma wants to determine the number of bacterial cells present after 24 hours (the 7th term of the sequence) and the total number of cells that have existed over the entire 24-hour period (sum of the first 7 terms). Find the number of cells at 24 hours and the total number of cells over the first 7 time intervals. Answer: a7 = 5103, S7 = 7651 Solution: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 7 (since we start at time 0 and have intervals every 4 hours up to 24 hours, that gives 7 terms).
    Full step-by-step solution

    Step 1: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 7 (since we start at time 0 and have intervals every 4 hours up to 24 hours, that gives 7 terms). Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=7, a7 = 7 * 3^(7-1) = 7 * 3^6. Step 3: Calculate 3^6. 3^2 = 9, 3^4 = 9 * 9 = 81, 3^6 = 81 * 9 = 729. Step 4: Multiply by 7: a7 = 7 * 729 = 5103. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=7, S7 = 7 * (1 - 3^7) / (1 - 3). Step 6: Calculate 3^7 = 3 * 3^6 = 3 * 729 = 2187. Step 7: Numerator: 1 - 2187 = -2186. Denominator: 1 - 3 = -2. Step 8: (1 - r^n) / (1 - r) = (-2186) / (-2) = 1093. Step 9: Multiply by a1: S7 = 7 * 1093 = 7651. The number of cells at 24 hours is 5103 and the total number of cells over the first 7 time intervals is 7651.

  2. Aroha is an environmental scientist monitoring the population of a rare bird species in a protected wetland. She observes that the population follows a geometric growth pattern due to successful conservation efforts. In the first year of her study, there are 9 birds. Each subsequent year, the population increases by a factor of 4 times the previous year's population. Aroha wants to predict the bird population after 10 years (the 10th term of the sequence) and the total number of birds that have lived in the wetland over the first 10 years. Find the population in the 10th year and the total population over the first 10 years. Answer: a10 = 2359296, S10 = 3145725 Solution: Identify the first term a1 = 9 and the common ratio r = 4. The number of terms n = 10. Use the nth term formula a_n = a1 * r^(n-1).
    Full step-by-step solution

    Step 1: Identify the first term a1 = 9 and the common ratio r = 4. The number of terms n = 10. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n = 10, a10 = 9 * 4^(10-1) = 9 * 4^9. Step 3: Calculate 4^9. 4^2 = 16, 4^4 = 16^2 = 256, 4^8 = 256^2 = 65536. Then 4^9 = 4 * 65536 = 262144. Step 4: Multiply by 9: a10 = 9 * 262144 = 2359296. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n = 10, S10 = 9 * (1 - 4^10) / (1 - 4). Step 6: Calculate 4^10 = 4 * 4^9 = 4 * 262144 = 1048576. Step 7: Numerator: 1 - 1048576 = -1048575. Denominator: 1 - 4 = -3. Step 8: (1 - r^n) / (1 - r) = (-1048575) / (-3) = 349525. Step 9: Multiply by a1: S10 = 9 * 349525 = 3145725. Step 10: Alternatively, using the equivalent formula S_n = a1 * (r^n - 1) / (r - 1): S10 = 9 * (1048576 - 1) / (4 - 1) = 9 * 1048575 / 3 = 9 * 349525 = 3145725. The bird population in the 10th year is 2359296 birds, and the total population over the first 10 years is 3145725 birds.

  3. Emma is a botanist studying the population growth of a rare fern species in a protected reserve. She notices that the number of ferns in a specific plot follows a geometric pattern. In the first year, there are 7 ferns. Each subsequent year, the number of ferns triples compared to the previous year. Emma needs to report the number of ferns after 9 years (the 9th term of the sequence) and the total number of ferns that have appeared over the first 9 years. Find the number of ferns in the 9th year and the total number of ferns over the first 9 years. Answer: a9 = 45927, S9 = 68887 Solution: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Use the nth term formula a_n = a1 * r^(n-1).
    Full step-by-step solution

    Step 1: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=9, a9 = 7 * 3^(9-1) = 7 * 3^8. Step 3: Calculate 3^8. 3^2 = 9, 3^4 = 9^2 = 81, 3^8 = 81^2 = 6561. Step 4: Multiply by 7: a9 = 7 * 6561 = 45927. Step 5: Use the sum formula S_n = a1 * (r^n - 1) / (r - 1). For n=9, S9 = 7 * (3^9 - 1) / (3 - 1). Step 6: Calculate 3^9 = 3 * 3^8 = 3 * 6561 = 19683. Step 7: Numerator: 19683 - 1 = 19682. Denominator: 3 - 1 = 2. Step 8: (r^n - 1)/(r - 1) = 19682 / 2 = 9841. Step 9: Multiply by a1: S9 = 7 * 9841 = 68887. The number of ferns in the 9th year is 45927 and the total number over the first 9 years is 68887.

  4. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration C(t) in mg/L follows the function C(t) = 80e^(-0.15t) + 20e^(-0.03t), where t is time in hours. The therapeutic window for this drug is between 40 mg/L and 85 mg/L. Determine the total time interval during which the drug concentration remains within the therapeutic window. Answer: Approximately 8.42 hours Solution: In pharmacokinetics, determining the duration of therapeutic effect involves finding when a drug concentration function crosses specific threshold values.
    Full step-by-step solution

    In pharmacokinetics, determining the duration of therapeutic effect involves finding when a drug concentration function crosses specific threshold values. For exponential decay models, this typically requires solving transcendental equations that combine multiple exponential terms. The solution process often involves numerical methods or algebraic manipulation to isolate the time variable. Understanding how to work with exponential functions and their properties is essential for modeling real-world biological processes like drug metabolism.

  5. Matiu is a marine biologist studying the population growth of a rare species of coral on a protected reef. He notes that the coral covers an area that grows geometrically each year. In the first year, the coral covers 24 square metres. Each subsequent year, the area covered increases by a factor of 2.5 times the previous year's area. Matiu needs to report the area covered after 10 years (the 10th term of the sequence) and the total area covered over the first 10 years for a conservation grant application. Find the area covered in the 10th year and the total area covered over the first 10 years. Answer: a10 = 366210.9375, S10 = 610351.5625 Solution: Identify the first term a1 = 24 and the common ratio r = 2.5. The number of terms n = 10. Use the nth term formula a_n = a1 * r^(n-1).
    Full step-by-step solution

    Step 1: Identify the first term a1 = 24 and the common ratio r = 2.5. The number of terms n = 10. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=10, a10 = 24 * (2.5)^(10-1) = 24 * (2.5)^9. Step 3: Calculate (2.5)^9. 2.5 = 5/2. (5/2)^9 = 5^9 / 2^9 = 1953125 / 512 = 3814.697265625. Step 4: Multiply by 24: a10 = 24 * 3814.697265625 = 91552.734375. (Wait, recalculate carefully: 1953125/512 = 3814.697265625, then 24 * 3814.697265625 = 91552.734375. But check: 24 * 1953125 = 46875000, divided by 512 = 46875000/512 = 91552.734375. However, this seems too large. Let's recalculate (2.5)^9 step by step: 2.5^2 = 6.25, 2.5^4 = 6.25^2 = 39.0625, 2.5^8 = 39.0625^2 = 1525.87890625, then 2.5^9 = 1525.87890625 * 2.5 = 3814.697265625. So a10 = 24 * 3814.697265625 = 91552.734375. That is correct.) Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=10, S10 = 24 * (1 - 2.5^10) / (1 - 2.5). Step 6: Calculate 2.5^10 = 2.5 * 2.5^9 = 2.5 * 3814.697265625 = 9536.7431640625. Step 7: Numerator: 1 - 9536.7431640625 = -9535.7431640625. Denominator: 1 - 2.5 = -1.5. Step 8: (1 - r^n)/(1 - r) = (-9535.7431640625)/(-1.5) = 6357.162109375. Step 9: Multiply by a1: S10 = 24 * 6357.162109375 = 152571.890625. Step 10: Verify using alternate formula S_n = a1*(r^n - 1)/(r - 1): S10 = 24 * (9536.7431640625 - 1)/(2.5 - 1) = 24 * 9535.7431640625/1.5 = 24 * 6357.162109375 = 152571.890625. The area in the 10th year is 91552.734375 square metres and the total area over 10 years is 152571.890625 square metres.