Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 1

1. 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.

2. Isabella is analyzing the population decline of an endangered species in a wildlife reserve. The population was 3,125 individuals in 2010, and it has been decreasing by 20% each year. She wants to determine the population in 2015 and the year when the population will drop below 1,000 individuals. Show that this situation represents a geometric sequence, identify the common ratio, and write the explicit formula for the population after n years (where n = 0 corresponds to 2010). Then use your formula to answer her questions. Answer: Population in 2015 is 1,024; population drops below 1,000 in 2016 Solution: Identify that the population decreases by 20% each year, meaning it retains 80% (or 0.80) of the previous year's population. This constant multiplier (r = 0.80) confirms a geometric sequence.
   Full step-by-step solution

   Step 1: Identify that the population decreases by 20% each year, meaning it retains 80% (or 0.80) of the previous year's population. This constant multiplier (r = 0.80) confirms a geometric sequence. Step 2: First term a_0 = 3125 (in 2010, n=0). Common ratio r = 0.80. Step 3: Explicit formula: P(n) = 3125 * (0.80)^n, where n is the number of years after 2010. Step 4: For 2015, n = 5. P(5) = 3125 * (0.80)^5 = 3125 * 0.32768 = 1024. Step 5: To find when population drops below 1000, solve 3125 * (0.80)^n < 1000. Step 6: Divide both sides by 3125: (0.80)^n < 1000/3125 = 0.32. Step 7: Take log base 0.80 of both sides (or use natural logs): n > log(0.32)/log(0.80). Step 8: log(0.32) ≈ -0.49485, log(0.80) ≈ -0.09691. So n > (-0.49485)/(-0.09691) ≈ 5.106. Step 9: Since n must be an integer year, n = 6. That corresponds to 2010 + 6 = 2016. Step 10: Verify: P(5) = 1024 (still above 1000), P(6) = 3125 * (0.80)^6 = 3125 * 0.262144 = 819.2 (below 1000). The answer is: Population in 2015 is 1024; population drops below 1000 in 2016.

3. Liam is a financial analyst studying the depreciation of a specialized piece of manufacturing equipment. The equipment's value (in thousands of dollars) at the end of each year forms a geometric sequence. After the first year, the equipment is worth $405,000. After the third year, it is worth $245,000. Assuming the common ratio is positive and less than 1, determine the explicit formula for the value of the equipment V(n) after n years, where V(n) is in thousands of dollars. Answer: V(n) = 405 * (7/9)^(n-1) Solution: Define the geometric sequence. Let V(1) = a be the value after 1 year. We are given V(1) = 405 thousand dollars.
   Full step-by-step solution

   Step 1: Define the geometric sequence. Let V(1) = a be the value after 1 year. We are given V(1) = 405 thousand dollars. Let the common ratio be r. Step 2: The general formula is V(n) = a * r^(n-1). Step 3: We know V(3) = 245 thousand dollars. Using the formula with n=3: V(3) = a * r^(2) = 405 * r^2. Step 4: Set up the equation: 405 * r^2 = 245. Step 5: Divide both sides by 405: r^2 = 245 / 405. Step 6: Simplify the fraction: Divide numerator and denominator by 5: 245/405 = 49/81. Step 7: Take the positive square root (since r is positive and less than 1): r = sqrt(49/81) = 7/9. Step 8: Write the explicit formula: V(n) = 405 * (7/9)^(n-1). The explicit formula is V(n) = 405 * (7/9)^(n-1).

4. 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.

5. Noah is analyzing a geometric sequence that models the depreciation of a rare car. The car's value in thousands of dollars after n years follows the sequence: 80, 60, 45, 33.75, ... Determine the common ratio r and write the explicit formula for the value V(n) after n years (with n starting at 0). Then, using that formula, find the value of the car after 5 years, rounded to the nearest thousand dollars. Answer: 19 Solution: Check if the sequence is geometric. Divide 60 by 80 = 0.75. Divide 45 by 60 = 0.75.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric. Divide 60 by 80 = 0.75. Divide 45 by 60 = 0.75. Divide 33.75 by 45 = 0.75. The common ratio r = 0.75. Step 2: The first term a_0 = 80 (thousand dollars). The explicit formula is V(n) = 80 * (0.75)^n. Step 3: To find the value after 5 years, substitute n=5: V(5) = 80 * (0.75)^5. Step 4: Calculate (0.75)^5 = 0.75 * 0.75 * 0.75 * 0.75 * 0.75 = 0.2373046875. Step 5: Multiply by 80: 80 * 0.2373046875 = 18.984375. Step 6: Round to the nearest thousand: 19 thousand dollars. The answer is 19.

6. Is 21, 126, 756, 4536... geometric? Find common ratio r and write the explicit formula aₙ. Answer: r = 6, aₙ = 21(6)ⁿ⁻¹ Solution: Check if the sequence is geometric by finding the ratio between consecutive terms. 126 ÷ 21 = 6 756 ÷ 126 = 6 4536 ÷ 756 = 6 Since all ratios equal 6, the sequence is geometric with common ratio r = 6.
   Full step-by-step solution

   Step 1: Check if the sequence is geometric by finding the ratio between consecutive terms. Step 2: 126 ÷ 21 = 6 Step 3: 756 ÷ 126 = 6 Step 4: 4536 ÷ 756 = 6 Step 5: Since all ratios equal 6, the sequence is geometric with common ratio r = 6. Step 6: The first term a₁ = 21. The explicit formula is aₙ = a₁(r)ⁿ⁻¹ = 21(6)ⁿ⁻¹. The common ratio is 6 and the explicit formula is aₙ = 21(6)ⁿ⁻¹.

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.