Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 3

1. Is 15, 45, 135, 405... geometric? Find common ratio Answer: 3 Solution: Check if the sequence is geometric by finding the ratio between consecutive terms. 45 ÷ 15 = 3 135 ÷ 45 = 3 405 ÷ 135 = 3 Since all ratios equal 3, the sequence is geometric with common ratio r = 3.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by finding the ratio between consecutive terms. Step 2: 45 ÷ 15 = 3 Step 3: 135 ÷ 45 = 3 Step 4: 405 ÷ 135 = 3 Step 5: Since all ratios equal 3, the sequence is geometric with common ratio r = 3. The common ratio is 3.

2. Hana is training for a marathon by following a recovery plan that reduces her weekly running distance by a constant percentage each week. In the first week, she runs 40 km. In the second week, she runs 32 km. In the third week, she runs 25.6 km. Determine the common ratio of this geometric sequence and write an explicit formula for the distance she runs in week n. Answer: r = 0.8; a_n = 40(0.8)^(n-1) Solution: Identify the first three terms: a_1 = 40, a_2 = 32, a_3 = 25.6. Find the ratio between consecutive terms: r = a_2 / a_1 = 32 / 40 = 0.8. So r = 0.8.
   Full step-by-step solution

   Step 1: Identify the first three terms: a_1 = 40, a_2 = 32, a_3 = 25.6. Step 2: Find the ratio between consecutive terms: r = a_2 / a_1 = 32 / 40 = 0.8. Step 3: Verify the ratio is constant: a_3 / a_2 = 25.6 / 32 = 0.8. So r = 0.8. Step 4: The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1). Step 5: Substitute a_1 = 40 and r = 0.8: a_n = 40(0.8)^(n-1). The answer is r = 0.8 and a_n = 40(0.8)^(n-1).

3. Is 5, 15, 45, 135... geometric? Find the common ratio r = ? Answer: 3 Solution: Check the ratio between the second and first term: 15 ÷ 5 = 3 Check the ratio between the third and second term: 45 ÷ 15 = 3 Check the ratio between the fourth and third term: 135 ÷ 45 = 3 Since the ratio is constant (3) between all consecutive terms, this is a geometric sequence.
   Full step-by-step solution

   Step 1: Check the ratio between the second and first term: 15 ÷ 5 = 3 Step 2: Check the ratio between the third and second term: 45 ÷ 15 = 3 Step 3: Check the ratio between the fourth and third term: 135 ÷ 45 = 3 Step 4: Since the ratio is constant (3) between all consecutive terms, this is a geometric sequence. Step 5: The common ratio r = 3.

4. Is 8, 40, 200, 1000... geometric? Find common ratio Answer: 5 Solution: 40 ÷ 8 = 5 200 ÷ 40 = 5 1000 ÷ 200 = 5 Since all ratios equal 5, this is a geometric sequence The common ratio is 5 The answer is 5.
   Full step-by-step solution

   Step 1: Check the ratio between consecutive terms 40 ÷ 8 = 5 200 ÷ 40 = 5 1000 ÷ 200 = 5 Step 2: Since all ratios equal 5, this is a geometric sequence Step 3: The common ratio is 5 The answer is 5.

5. Mason is investigating a geometric sequence that models the number of views a viral video receives each hour. The first term of the sequence is 12, and the fourth term is 324. Determine the common ratio of this geometric sequence, and write the explicit formula for the nth term. Answer: r = 3; a_n = 12 * 3^(n-1) Solution: The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1). Given a_1 = 12 and a_4 = 324, substitute n = 4: 324 = 12 * r^(4-1) = 12 * r^3. Divide both sides by 12: r^3 = 324 / 12 = 27.
   Full step-by-step solution

   Step 1: The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1). Step 2: Given a_1 = 12 and a_4 = 324, substitute n = 4: 324 = 12 * r^(4-1) = 12 * r^3. Step 3: Divide both sides by 12: r^3 = 324 / 12 = 27. Step 4: Take the cube root of both sides: r = cube root of 27 = 3. Step 5: The common ratio is r = 3. Step 6: Write the explicit formula: a_n = 12 * 3^(n-1). The answer is r = 3 and a_n = 12 * 3^(n-1).

6. 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.