Geometric Sequences

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Answer Key & Explanations

Geometric Sequences · Grade 12 · Worksheet 1

1. Isabella is a financial analyst for a renewable energy company. The company's revenue has been growing geometrically due to increased solar panel installations. In the first year of tracking, the revenue was $8,000. Each subsequent year, the revenue multiplies by a factor of 2.5. Isabella needs to present the revenue projections for the 7th year and the total cumulative revenue over the first 7 years to the board of directors. Determine the revenue in the 7th year and the total revenue over the first 7 years. Answer: a7 = $781,250, S7 = $1,302,080 Solution: Identify the parameters. The first term a1 = 8000. The common ratio r = 2.5.
   Full step-by-step solution

   Step 1: Identify the parameters. The first term a1 = 8000. The common ratio r = 2.5. The number of terms n = 7. Step 2: Find the 7th term (revenue in the 7th year) using the formula a_n = a1 * r^(n-1). a7 = 8000 * (2.5)^(7-1) = 8000 * (2.5)^6 Step 3: Calculate (2.5)^6. 2.5 = 5/2. (5/2)^6 = 15625/64 = 244.140625. Step 4: Multiply by a1: a7 = 8000 * 244.140625 = 1,953,125. (Correction: 8000 * 244.140625 = 1,953,125. Let's recalculate carefully: (2.5)^2 = 6.25, (2.5)^4 = 6.25^2 = 39.0625, (2.5)^6 = 39.0625 * 6.25 = 244.140625. Then 8000 * 244.140625 = 1,953,125.) Step 5: Find the sum of the first 7 terms using the formula S_n = a1 * (r^n - 1) / (r - 1). S7 = 8000 * ( (2.5)^7 - 1 ) / (2.5 - 1) Step 6: Calculate (2.5)^7 = (2.5)^6 * 2.5 = 244.140625 * 2.5 = 610.3515625. Step 7: Substitute into the sum formula. S7 = 8000 * (610.3515625 - 1) / (1.5) S7 = 8000 * (609.3515625) / 1.5 Step 8: Divide the numerator by the denominator: 609.3515625 / 1.5 = 406.234375. Step 9: Multiply by a1: S7 = 8000 * 406.234375 = 3,249,875. (Correction: 8000 * 406.234375 = 3,249,875. Let's use fractions to be precise. r = 5/2. r^7 = (5/2)^7 = 78125/128. r^7 - 1 = (78125/128) - (128/128) = 77997/128. r - 1 = 3/2. (r^7 - 1)/(r - 1) = (77997/128) / (3/2) = (77997/128) * (2/3) = (77997 * 2) / (128 * 3) = 155994 / 384 = 25999/64. S7 = 8000 * (25999/64) = (8000/64) * 25999 = 125 * 25999 = 3,249,875.) The revenue in the 7th year is $1,953,125 and the total revenue over the first 7 years is $3,249,875.

2. Aroha is an environmental scientist monitoring the population of a rare bird species in a protected wetland. She observes that the population follows a geometric growth pattern due to successful conservation efforts. In the first year of her study, there are 9 birds. Each subsequent year, the population increases by a factor of 4 times the previous year's population. Aroha wants to predict the bird population after 10 years (the 10th term of the sequence) and the total number of birds that have lived in the wetland over the first 10 years. Find the population in the 10th year and the total population over the first 10 years. Answer: a10 = 2359296, S10 = 3145725 Solution: Identify the first term a1 = 9 and the common ratio r = 4. The number of terms n = 10. Use the nth term formula a_n = a1 * r^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 9 and the common ratio r = 4. The number of terms n = 10. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n = 10, a10 = 9 * 4^(10-1) = 9 * 4^9. Step 3: Calculate 4^9. 4^2 = 16, 4^4 = 16^2 = 256, 4^8 = 256^2 = 65536. Then 4^9 = 4 * 65536 = 262144. Step 4: Multiply by 9: a10 = 9 * 262144 = 2359296. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n = 10, S10 = 9 * (1 - 4^10) / (1 - 4). Step 6: Calculate 4^10 = 4 * 4^9 = 4 * 262144 = 1048576. Step 7: Numerator: 1 - 1048576 = -1048575. Denominator: 1 - 4 = -3. Step 8: (1 - r^n) / (1 - r) = (-1048575) / (-3) = 349525. Step 9: Multiply by a1: S10 = 9 * 349525 = 3145725. Step 10: Alternatively, using the equivalent formula S_n = a1 * (r^n - 1) / (r - 1): S10 = 9 * (1048576 - 1) / (4 - 1) = 9 * 1048575 / 3 = 9 * 349525 = 3145725. The bird population in the 10th year is 2359296 birds, and the total population over the first 10 years is 3145725 birds.

3. Emma is a botanist studying the population growth of a rare fern species in a protected reserve. She notices that the number of ferns in a specific plot follows a geometric pattern. In the first year, there are 7 ferns. Each subsequent year, the number of ferns triples compared to the previous year. Emma needs to report the number of ferns after 9 years (the 9th term of the sequence) and the total number of ferns that have appeared over the first 9 years. Find the number of ferns in the 9th year and the total number of ferns over the first 9 years. Answer: a9 = 45927, S9 = 68887 Solution: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Use the nth term formula a_n = a1 * r^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=9, a9 = 7 * 3^(9-1) = 7 * 3^8. Step 3: Calculate 3^8. 3^2 = 9, 3^4 = 9^2 = 81, 3^8 = 81^2 = 6561. Step 4: Multiply by 7: a9 = 7 * 6561 = 45927. Step 5: Use the sum formula S_n = a1 * (r^n - 1) / (r - 1). For n=9, S9 = 7 * (3^9 - 1) / (3 - 1). Step 6: Calculate 3^9 = 3 * 3^8 = 3 * 6561 = 19683. Step 7: Numerator: 19683 - 1 = 19682. Denominator: 3 - 1 = 2. Step 8: (r^n - 1)/(r - 1) = 19682 / 2 = 9841. Step 9: Multiply by a1: S9 = 7 * 9841 = 68887. The number of ferns in the 9th year is 45927 and the total number over the first 9 years is 68887.

4. Aroha is a marine biologist studying the population of a rare species of sea turtle. She observes that the number of nesting females on a protected beach follows a geometric pattern. In the first year of her study, there are 7 nesting females. Each subsequent year, the number of nesting females increases by a factor of 3 times the previous year's number. Aroha wants to predict the number of nesting females in the 9th year and the total number of nesting females over the first 9 years. Find the number of nesting females in the 9th year and the total number over the first 9 years. Answer: a9 = 45927, S9 = 68887 Solution: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Use the nth term formula a_n = a1 * r^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 9. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=9, a9 = 7 * 3^(9-1) = 7 * 3^8. Step 3: Calculate 3^8. 3^2 = 9, 3^4 = 9^2 = 81, 3^8 = 81^2 = 6561. Step 4: Multiply by 7: a9 = 7 * 6561 = 45927. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=9, S9 = 7 * (1 - 3^9) / (1 - 3). Step 6: Calculate 3^9 = 3 * 3^8 = 3 * 6561 = 19683. Step 7: Numerator: 1 - 19683 = -19682. Denominator: 1 - 3 = -2. Step 8: (1 - r^n)/(1 - r) = (-19682)/(-2) = 9841. Step 9: Multiply by a1: S9 = 7 * 9841 = 68887. Step 10: Verify: Alternatively, sum = a1 * (r^n - 1)/(r - 1) = 7 * (19683 - 1)/(2) = 7 * 19682/2 = 7 * 9841 = 68887. The number of nesting females in the 9th year is 45927 and the total number over the first 9 years is 68887.

5. ∑(n=1 to 6) 5 × (-4)^(n-1) = ? Answer: -4095 Solution: a₁ = 5 × (-4)^(1-1) = 5 × (-4)^0 = 5 × 1 = 5 r = -4 Sₙ = a₁ × (1 - rⁿ) / (1 - r) Substitute n = 6, a₁ = 5, r = -4 S₆ = 5 × (1 - (-4)⁶) / (1 - (-4)) Calculate (-4)⁶ (-4)⁶ = (-4) × (-4) × (-4) × (-4) × (-4) × (-4) = 16 × 16 × 16 = 256 × 16 = 4096 S₆ = 5 × (1 - 4096) / (1 + 4) S₆ = 5 × (-4095) / 5…
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and common ratio (r) a₁ = 5 × (-4)^(1-1) = 5 × (-4)^0 = 5 × 1 = 5 r = -4 Step 2: Use the geometric series sum formula for n terms Sₙ = a₁ × (1 - rⁿ) / (1 - r) Step 3: Substitute n = 6, a₁ = 5, r = -4 S₆ = 5 × (1 - (-4)⁶) / (1 - (-4)) Step 4: Calculate (-4)⁶ (-4)⁶ = (-4) × (-4) × (-4) × (-4) × (-4) × (-4) = 16 × 16 × 16 = 256 × 16 = 4096 Step 5: Substitute into the formula S₆ = 5 × (1 - 4096) / (1 + 4) S₆ = 5 × (-4095) / 5 S₆ = 5 × (-819) S₆ = -4095 Step 6: Verify by calculating individual terms Term 1: 5 × (-4)^0 = 5 Term 2: 5 × (-4)^1 = -20 Term 3: 5 × (-4)^2 = 80 Term 4: 5 × (-4)^3 = -320 Term 5: 5 × (-4)^4 = 1280 Term 6: 5 × (-4)^5 = -5120 Sum = 5 + (-20) + 80 + (-320) + 1280 + (-5120) = -15 + 80 + (-320) + 1280 + (-5120) = 65 + (-320) + 1280 + (-5120) = -255 + 1280 + (-5120) = 1025 + (-5120) = -4095 The answer is -4095.