Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 1

1. Is 4, 12, 36, 108... geometric? Find common ratio Answer: 3 Solution: 12 ÷ 4 = 3 36 ÷ 12 = 3 108 ÷ 36 = 3 Since all ratios equal 3, this is a geometric sequence The common ratio r = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Check the ratio between consecutive terms 12 ÷ 4 = 3 36 ÷ 12 = 3 108 ÷ 36 = 3 Step 2: Since all ratios equal 3, this is a geometric sequence Step 3: The common ratio r = 3 The answer is 3.

2. Sophia is an architect designing a modern art installation. The structure consists of a series of triangular panels arranged in a vertical tower. The area of the first panel is 192 square centimeters. Each subsequent panel's area is 3/4 of the area of the panel directly below it. If the pattern continues indefinitely, what is the total area of all the panels in the infinite tower? Answer: 768 Solution: Identify the first term. The area of the first panel is a1 = 192 square centimeters. Identify the common ratio.
   Full step-by-step solution

   Step 1: Identify the first term. The area of the first panel is a1 = 192 square centimeters. Step 2: Identify the common ratio. Each panel's area is 3/4 of the previous, so r = 3/4 = 0.75. Step 3: Since |r| < 1, the infinite geometric series converges. Use the formula for the sum of an infinite geometric series: S = a1 / (1 - r). Step 4: Substitute the values: S = 192 / (1 - 3/4). Step 5: Simplify the denominator: 1 - 3/4 = 1/4. Step 6: Divide: S = 192 / (1/4) = 192 * 4 = 768. Step 7: The total area of all panels is 768 square centimeters.

3. Charlotte is a microbiologist studying the growth of a bacterial colony in a petri dish. She observes that the colony's area increases by a factor of 1.25 every day. On the first day, the area is 16 cm². She wants to predict the area on the nth day. Identify the type of sequence, find the common ratio, and write the explicit formula for the area Aₙ in cm² on day n. Answer: Geometric sequence, r = 1.25, Aₙ = 16 × (1.25)^(n-1) Solution: Check if the sequence is geometric by finding the ratio between consecutive terms. The area increases by a factor of 1.25 each day, so the ratio is constant at 1.25. This confirms it is a geometric sequence.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by finding the ratio between consecutive terms. The area increases by a factor of 1.25 each day, so the ratio is constant at 1.25. This confirms it is a geometric sequence. Step 2: Identify the first term a₁ = 16 cm² (area on day 1). Step 3: Identify the common ratio r = 1.25. Step 4: The explicit formula for a geometric sequence is Aₙ = a₁ × r^(n-1). Step 5: Substitute the values: Aₙ = 16 × (1.25)^(n-1). The answer is: Geometric sequence, r = 1.25, Aₙ = 16 × (1.25)^(n-1).

4. Charlotte is a computer scientist studying the data storage efficiency of a new compression algorithm. She observes that the algorithm processes data in a pattern where the number of data points processed in each successive minute forms a geometric sequence. In the first minute, 12 data points are processed. In the second minute, 48 data points are processed. In the third minute, 192 data points are processed. If this pattern continues, what is the explicit formula for the number of data points processed in the nth minute? Answer: a_n = 12 * 4^(n-1) Solution: Identify the first term. The first term a_1 = 12 (data points processed in the first minute). Find the common ratio r.
   Full step-by-step solution

   Step 1: Identify the first term. The first term a_1 = 12 (data points processed in the first minute). Step 2: Find the common ratio r. Divide the second term by the first term: 48 / 12 = 4. Divide the third term by the second term: 192 / 48 = 4. Since the ratio is constant, r = 4. Step 3: Use the explicit formula for a geometric sequence: a_n = a_1 * r^(n-1). Step 4: Substitute the known values: a_1 = 12 and r = 4. Step 5: The explicit formula is a_n = 12 * 4^(n-1). The answer is a_n = 12 * 4^(n-1).

5. Charlotte is studying the growth of a certain species of bamboo. She measures the height of a new shoot and records the following heights at the end of each week: Week 1: 24 cm, Week 2: 36 cm, Week 3: 54 cm, Week 4: 81 cm. Assuming the height follows a geometric sequence, determine the common ratio and write an explicit formula for the height aₙ (in cm) at the end of week n. Answer: r = 1.5, aₙ = 24(1.5)^(n-1) Solution: Check for constant ratio: 36/24 = 1.5, 54/36 = 1.5, 81/54 = 1.5. Since all ratios are equal to 1.5, the sequence is geometric with common ratio r = 1.5. The first term a₁ = 24 cm.
   Full step-by-step solution

   Step 1: Check for constant ratio: 36/24 = 1.5, 54/36 = 1.5, 81/54 = 1.5. Since all ratios are equal to 1.5, the sequence is geometric with common ratio r = 1.5. Step 2: The first term a₁ = 24 cm. Step 3: The explicit formula for a geometric sequence is aₙ = a₁ * r^(n-1). Step 4: Substitute a₁ = 24 and r = 1.5: aₙ = 24 * (1.5)^(n-1). Thus, the common ratio is 1.5 and the explicit formula is aₙ = 24(1.5)^(n-1).

6. Is 9, 63, 441, 3087... geometric? Find common ratio r = ? Answer: 7 Solution: Check the ratio between the second and first term: 63 ÷ 9 = 7. Check the ratio between the third and second term: 441 ÷ 63 = 7. Check the ratio between the fourth and third term: 3087 ÷ 441 = 7.
   Full step-by-step solution

   Step 1: Check the ratio between the second and first term: 63 ÷ 9 = 7. Step 2: Check the ratio between the third and second term: 441 ÷ 63 = 7. Step 3: Check the ratio between the fourth and third term: 3087 ÷ 441 = 7. Step 4: Since the ratio is constant (7) for all consecutive terms, the sequence is geometric. Step 5: The common ratio r is 7.

7. A geometric sequence has first term 3 and common ratio 2. The sum of the first n terms is 3069. Find the value of n. Answer: 10 Solution: We are given a geometric sequence with first term a = 3 and common ratio r = 2. The sum of the first n terms is S_n = 3069. Write the formula for the sum of the first n terms of a geometric sequence.
   Full step-by-step solution

   We are given a geometric sequence with first term a = 3 and common ratio r = 2. The sum of the first n terms is S_n = 3069. --- **Step 1: Write the formula for the sum of the first n terms of a geometric sequence.** The sum is: S_n = a * (r^n - 1) / (r - 1) when r ≠ 1. Here a = 3, r = 2, S_n = 3069. --- **Step 2: Substitute the known values into the formula.** 3069 = 3 * (2^n - 1) / (2 - 1) Since 2 - 1 = 1, the denominator is 1, so: 3069 = 3 * (2^n - 1) --- **Step 3: Solve for 2^n.** Divide both sides by 3: 3069 / 3 = 2^n - 1 3069 ÷ 3 = 1023 So: 1023 = 2^n - 1 Add 1 to both sides: 1023 + 1 = 2^n 1024 = 2^n --- **Step 4: Recognize 1024 as a power of 2.** 1024 = 2^10 So: 2^n = 2^10 Thus n = 10. --- **Step 5: Conclusion** The value of n is 10.