Geometric Sequences

lessonbunny.com

Answer Key & Explanations

Geometric Sequences · Grade 12 · Worksheet 2

1. Olivia is an environmental scientist monitoring the population of a certain fish species in a lake. She observes that the population decreases by 20% each year due to environmental changes. The initial population is 10,000 fish. She wants to find the number of fish after n years. Is this a geometric sequence? If so, find the common ratio and write the explicit formula for the population after n years. Answer: Yes, common ratio = 0.8, a_n = 10000 * (0.8)^(n-1) Solution: Identify if the pattern is geometric. The population decreases by 20% each year, meaning each year the population is multiplied by (1 - 0.20) = 0.8. This is a constant ratio, so it is a geometric sequence.
   Full step-by-step solution

   Step 1: Identify if the pattern is geometric. The population decreases by 20% each year, meaning each year the population is multiplied by (1 - 0.20) = 0.8. This is a constant ratio, so it is a geometric sequence. Step 2: Find the common ratio r. r = 0.8. Step 3: Identify the first term a_1. a_1 = 10,000. Step 4: Write the explicit formula for a geometric sequence: a_n = a_1 * r^(n-1). Step 5: Substitute the values: a_n = 10000 * (0.8)^(n-1). The answer is: Yes, common ratio = 0.8, a_n = 10000 * (0.8)^(n-1).

2. Tane is studying a fractal pattern made of circles. The largest circle has a radius of 27 cm. A second circle is drawn inside the first, tangent to it at a single point, with its center on a diameter of the larger circle. The radius of each subsequent circle is exactly one-third the radius of the previous circle. This process of drawing smaller tangent circles continues infinitely along the same diameter. What is the total length of all the diameters of every circle in this infinite pattern? Answer: 81 cm Solution: The first circle has radius 27 cm, so its diameter is 2 * 27 = 54 cm. The second circle has radius 27 * (1/3) = 9 cm, so its diameter is 2 * 9 = 18 cm.
   Full step-by-step solution

   Step 1: The first circle has radius 27 cm, so its diameter is 2 * 27 = 54 cm. Step 2: The second circle has radius 27 * (1/3) = 9 cm, so its diameter is 2 * 9 = 18 cm. Step 3: The third circle has radius 9 * (1/3) = 3 cm, so its diameter is 2 * 3 = 6 cm. Step 4: The diameters form the geometric sequence: 54, 18, 6, 2, ... Step 5: The common ratio r is 18 / 54 = 1/3. Check: 6 / 18 = 1/3, 2 / 6 = 1/3. Yes, it is constant. Step 6: This is an infinite geometric series. The first term a = 54, and the common ratio r = 1/3. Since |r| < 1, the sum converges. Step 7: The formula for the sum of an infinite geometric series is S = a / (1 - r). Step 8: Substitute the values: S = 54 / (1 - 1/3) = 54 / (2/3) = 54 * (3/2) = 81. Step 9: The total length of all the diameters is 81 cm.

3. Is 7, 28, 112, 448... geometric? Find common ratio = ? Answer: 4 Solution: 28 ÷ 7 = 4 112 ÷ 28 = 4 448 ÷ 112 = 4 Since all ratios equal 4, the sequence is geometric with common ratio 4 The answer is 4.
   Full step-by-step solution

   Step 1: Check the ratio between consecutive terms Step 2: 28 ÷ 7 = 4 Step 3: 112 ÷ 28 = 4 Step 4: 448 ÷ 112 = 4 Step 5: Since all ratios equal 4, the sequence is geometric with common ratio 4 The answer is 4.

4. Isabella is studying a chain reaction in her chemistry class. She observes that the number of molecules involved in the reaction forms a sequence: 7, 42, 252, 1512, ... Determine whether this sequence is geometric. If it is, find the common ratio and write the explicit formula for the nth term a_n. Answer: Yes, geometric; r = 6; a_n = 7 * 6^(n-1) Solution: Check if the sequence is geometric by dividing consecutive terms. 42 / 7 = 6 252 / 42 = 6 1512 / 252 = 6 The ratio is constant (6), so the sequence is geometric. Identify the first term a1 = 7 and common ratio r = 6.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by dividing consecutive terms. 42 / 7 = 6 252 / 42 = 6 1512 / 252 = 6 The ratio is constant (6), so the sequence is geometric. Step 2: Identify the first term a1 = 7 and common ratio r = 6. Step 3: Write the explicit formula for a geometric sequence: a_n = a1 * r^(n-1). Substitute a1 = 7 and r = 6: a_n = 7 * 6^(n-1). The answer is: Yes, it is geometric; r = 6; a_n = 7 * 6^(n-1).

5. Is 16, 41, 106, 276... geometric? Find the common ratio if it is geometric, or write 'no' if it is not. Answer: A. no Solution: Calculate the ratio between the second and first terms: 41 / 16 = 2.5625 Calculate the ratio between the third and second terms: 106 / 41 ≈ 2.5854 Calculate the ratio between the fourth and third terms: 276 / 106 ≈ 2.6038 Compare the ratios: 2.5625, 2.5854, and 2.6038 are not equal.
   Full step-by-step solution

   Step 1: Calculate the ratio between the second and first terms: 41 / 16 = 2.5625 Step 2: Calculate the ratio between the third and second terms: 106 / 41 ≈ 2.5854 Step 3: Calculate the ratio between the fourth and third terms: 276 / 106 ≈ 2.6038 Step 4: Compare the ratios: 2.5625, 2.5854, and 2.6038 are not equal. Step 5: Since the ratios are not constant, the sequence is not geometric. The answer is no.

6. Is 11, 55, 275, 1375... geometric? Find common ratio r = ? Answer: 5 Solution: Check if the sequence is geometric by finding the ratio between consecutive terms. Divide the second term by the first: 55 ÷ 11 = 5. Divide the third term by the second: 275 ÷ 55 = 5.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by finding the ratio between consecutive terms. Step 2: Divide the second term by the first: 55 ÷ 11 = 5. Step 3: Divide the third term by the second: 275 ÷ 55 = 5. Step 4: Divide the fourth term by the third: 1375 ÷ 275 = 5. Step 5: Since all ratios equal 5, the sequence is geometric with common ratio r = 5. The answer is 5.