Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 2

1. Matiu is a biologist studying the spread of a fungal infection in a native forest. He observes that the area covered by the fungus doubles every 12 days. Initially, the fungus covers 15 square meters. Matiu wants to predict the area covered after 60 days using a geometric sequence model. Write the explicit formula for the area covered after n days, where n is a multiple of 12, and then use it to find the area after 60 days. Answer: 480 Solution: Identify the first term (area at day 0): a₁ = 15 square meters. Identify the common ratio: the area doubles every 12 days, so r = 2.
   Full step-by-step solution

   Step 1: Identify the first term (area at day 0): a₁ = 15 square meters. Step 2: Identify the common ratio: the area doubles every 12 days, so r = 2. Step 3: Since the doubling happens every 12 days, the number of 12-day periods in n days is n/12. The explicit formula for the area A(n) after n days (where n is a multiple of 12) is: A(n) = a₁ * r^(n/12) = 15 * 2^(n/12). Step 4: Find the area after 60 days: substitute n = 60 into the formula: A(60) = 15 * 2^(60/12) = 15 * 2^5. Step 5: Calculate 2^5 = 32. Step 6: Multiply: 15 * 32 = 480. The area covered after 60 days is 480 square meters.

2. Is 12, 36, 108, 324... geometric? Find common ratio Answer: 3 Solution: Check if the sequence is geometric by finding the ratio between consecutive terms 36 ÷ 12 = 3 108 ÷ 36 = 3 324 ÷ 108 = 3 All ratios equal 3, so the sequence is geometric with common ratio 3 The common ratio is 3.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by finding the ratio between consecutive terms Step 2: 36 ÷ 12 = 3 Step 3: 108 ÷ 36 = 3 Step 4: 324 ÷ 108 = 3 Step 5: All ratios equal 3, so the sequence is geometric with common ratio 3 The common ratio is 3.

3. Matiu is studying a chain of interconnected pendulums in his physics class. The length of the first pendulum is 160 cm. Each subsequent pendulum in the chain has a length that is 85% of the length of the previous pendulum. If the pattern continues indefinitely, what is the total length of all the pendulums in the infinite chain? Answer: 1066.67 Solution: Identify the first term: a₁ = 160 cm. Identify the common ratio: r = 85% = 0.85. Since |r| = 0.85 < 1, the infinite geometric series converges.
   Full step-by-step solution

   Step 1: Identify the first term: a₁ = 160 cm. Step 2: Identify the common ratio: r = 85% = 0.85. Step 3: Since |r| = 0.85 < 1, the infinite geometric series converges. Step 4: Use the infinite geometric series formula: S = a₁ / (1 - r). Step 5: Substitute the values: S = 160 / (1 - 0.85). Step 6: Calculate denominator: 1 - 0.85 = 0.15. Step 7: Divide: 160 / 0.15 = 1066.666... Step 8: Round to two decimal places: S ≈ 1066.67 cm. The answer is 1066.67.

4. Aroha's geometric sequence: 7, 35, 175, 875... Write the explicit formula aₙ = ? Answer: 7×5^(n-1) Solution: Step 1: Identify the first term: a₁ = 7 Step 2: Find the common ratio: r = 35 ÷ 7 = 5 Step 3: Verify with next terms: 175 ÷ 35 = 5, 875 ÷ 175 = 5 Step 4: Write the explicit formula: aₙ = a₁ × r^(n-1) Step 5: Substitute values: aₙ = 7 × 5^(n-1) The explicit formula is aₙ = 7 × 5^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term: a₁ = 7 Step 2: Find the common ratio: r = 35 ÷ 7 = 5 Step 3: Verify with next terms: 175 ÷ 35 = 5, 875 ÷ 175 = 5 Step 4: Write the explicit formula: aₙ = a₁ × r^(n-1) Step 5: Substitute values: aₙ = 7 × 5^(n-1) The explicit formula is aₙ = 7 × 5^(n-1).

5. Emma is analyzing the decay of a radioactive isotope in her physics lab. The remaining mass M(t) in grams after t years is given by M(t) = 120 * e^(-0.04t). She needs to determine how long it will take for exactly half of the original mass to decay. How many years will this take? Round your answer to the nearest tenth of a year. Answer: 17.3 Solution: The original mass is M(0) = 120 * e^(-0.04*0) = 120 grams. When half has decayed, half remains, so M(t) = 120/2 = 60 grams.
   Full step-by-step solution

   Step 1: The original mass is M(0) = 120 * e^(-0.04*0) = 120 grams. Step 2: When half has decayed, half remains, so M(t) = 120/2 = 60 grams. Step 3: Set up the equation: 120 * e^(-0.04t) = 60 Step 4: Divide both sides by 120: e^(-0.04t) = 0.5 Step 5: Take natural logarithm of both sides: ln(e^(-0.04t)) = ln(0.5) Step 6: Simplify: -0.04t = ln(0.5) Step 7: Calculate ln(0.5) ≈ -0.693147 Step 8: Solve for t: t = -0.693147 / -0.04 = 17.328675 Step 9: Round to nearest tenth: t ≈ 17.3 years The answer is 17.3 years.

6. Olivia is a marine biologist studying the population of a rare species of sea star in a protected reef. She observes that the population follows a geometric sequence over consecutive years. In the first year of her study, she counts 81 sea stars. In the third year, she counts 729 sea stars. Assuming the population continues to grow at the same constant ratio each year, what will be the population in the fifth year? Answer: 6561 Solution: The first term a_1 = 81. The third term a_3 = 729. Using the explicit formula a_n = a_1 * r^(n-1), for n=3: 729 = 81 * r^(3-1) = 81 * r^2.
   Full step-by-step solution

   Step 1: The first term a_1 = 81. The third term a_3 = 729. Step 2: Using the explicit formula a_n = a_1 * r^(n-1), for n=3: 729 = 81 * r^(3-1) = 81 * r^2. Step 3: Divide both sides by 81: r^2 = 729 / 81 = 9. Step 4: Take the positive square root (since population grows): r = sqrt(9) = 3. Step 5: Now find the fifth term a_5: a_5 = 81 * 3^(5-1) = 81 * 3^4. Step 6: Calculate 3^4 = 81. Then a_5 = 81 * 81 = 6561. The answer is 6561 sea stars.

7. Isabella's geometric sequence: 7, 21, 63, 189... Find the common ratio r = ? Answer: 3 Solution: Step 1: Identify the first two terms: 7 and 21 Step 2: Calculate the ratio: 21 ÷ 7 = 3 Step 3: Verify with the next terms: 63 ÷ 21 = 3, 189 ÷ 63 = 3 Step 4: Since all ratios equal 3, the common ratio is 3 The answer is 3.
   Full step-by-step solution

   Step 1: Identify the first two terms: 7 and 21 Step 2: Calculate the ratio: 21 ÷ 7 = 3 Step 3: Verify with the next terms: 63 ÷ 21 = 3, 189 ÷ 63 = 3 Step 4: Since all ratios equal 3, the common ratio is 3 The answer is 3.