Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 2

1. Isabella is constructing a visual pattern using a series of nested regular heptagons. The outermost heptagon has a side length of 42 cm. Each subsequent heptagon is formed by connecting the midpoints of the sides of the previous heptagon, creating a smaller rotated heptagon inside. The side lengths of these heptagons follow a geometric sequence with a common ratio of 0.7. What is the side length of the 7th heptagon, and what is the sum of the perimeters of the first 7 heptagons? (Perimeter of a regular heptagon = 7 × side length) Answer: Side length of 7th heptagon = 42 × (0.7)^6 = 42 × 0.117649 = 4.941258 cm; Sum of perimeters = 294 × (1 - (0.7)^7) / (1 - 0.7) = 294 × (1 - 0.0823543) / 0.3 = 294 × 0.9176457 / 0.3 = 294 × 3.058819 = 899.3 cm (approximately) Solution: Identify the first term and common ratio. First term (side length of outermost heptagon): a1 = 42 cm Common ratio: r = 0.7 Find the side length of the 7th heptagon.
   Full step-by-step solution

   Step 1: Identify the first term and common ratio. First term (side length of outermost heptagon): a1 = 42 cm Common ratio: r = 0.7 Step 2: Find the side length of the 7th heptagon. Using the formula a_n = a1 * r^(n-1): a7 = 42 * (0.7)^(7-1) = 42 * (0.7)^6 Calculate (0.7)^6: 0.7^2 = 0.49 0.7^3 = 0.49 * 0.7 = 0.343 0.7^4 = 0.343 * 0.7 = 0.2401 0.7^5 = 0.2401 * 0.7 = 0.16807 0.7^6 = 0.16807 * 0.7 = 0.117649 a7 = 42 * 0.117649 = 4.941258 cm Step 3: Find the sum of the perimeters of the first 7 heptagons. Perimeter of first heptagon: P1 = 7 * 42 = 294 cm The perimeters form a geometric sequence with first term 294 and common ratio r = 0.7. Using the sum formula S_n = a1 * (1 - r^n) / (1 - r): S7 = 294 * (1 - (0.7)^7) / (1 - 0.7) First calculate (0.7)^7: 0.7^7 = 0.7^6 * 0.7 = 0.117649 * 0.7 = 0.0823543 Now: 1 - (0.7)^7 = 1 - 0.0823543 = 0.9176457 1 - 0.7 = 0.3 S7 = 294 * (0.9176457) / 0.3 = 294 * 3.058819 = 899.292786 cm Final answer: Side length of 7th heptagon = 4.941258 cm; Sum of perimeters of first 7 heptagons = 899.3 cm (approximately).

2. Olivia is a financial analyst studying the growth of a company's quarterly revenue. The revenue in the first quarter was $5000. The company projects that each subsequent quarter, the revenue will be 1.2 times the previous quarter's revenue. Olivia needs to present the projected revenue for the 10th quarter and the total projected revenue over the first 10 quarters to the board. Find the revenue in the 10th quarter and the total revenue over the first 10 quarters. Answer: a10 = $25,790.00, S10 = $129,791.00 Solution: Identify the first term a1 = 5000 and the common ratio r = 1.2. 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 = 5000 and the common ratio r = 1.2. The number of terms n = 10. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=10, a10 = 5000 * (1.2)^(10-1) = 5000 * (1.2)^9. Step 3: Calculate (1.2)^9. First, 1.2^2 = 1.44, 1.2^4 = 1.44^2 = 2.0736, 1.2^8 = 2.0736^2 = 4.29981696, then multiply by 1.2 to get 1.2^9 = 4.29981696 * 1.2 = 5.159780352. Step 4: Multiply by a1: a10 = 5000 * 5.159780352 = 25798.90176. Rounding to the nearest dollar gives $25,799. However, for exact calculation, keep as 25798.90176. But let's recalculate more precisely: (1.2)^9 = (6/5)^9 = 6^9 / 5^9 = 10077696 / 1953125 = 5.159780352. Then a10 = 5000 * 10077696 / 1953125 = (5000 * 10077696) / 1953125 = 50388480000 / 1953125 = 25798.90176. So a10 = $25,798.90 (to nearest cent). Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=10, S10 = 5000 * (1 - 1.2^10) / (1 - 1.2). Step 6: Calculate 1.2^10 = 1.2 * 1.2^9 = 1.2 * 5.159780352 = 6.1917364224. Step 7: Numerator: 1 - 6.1917364224 = -5.1917364224. Denominator: 1 - 1.2 = -0.2. Step 8: (1 - r^n)/(1 - r) = (-5.1917364224)/(-0.2) = 25.958682112. Step 9: Multiply by a1: S10 = 5000 * 25.958682112 = 129793.41056. Step 10: The revenue in the 10th quarter is $25,798.90 and the total revenue over 10 quarters is $129,793.41.

3. Mere is a materials engineer testing the decay of a rare metal's conductivity under repeated stress cycles. She observes that the conductivity (in Siemens per metre) follows a geometric sequence. The initial conductivity is 64 S/m. After each stress cycle, the conductivity decreases to 3/4 of its previous value. Mere needs to report the conductivity after the 6th cycle (the 6th term) and the total conductivity loss (the sum of the first 6 terms) for a safety analysis. Find the conductivity after the 6th cycle and the sum of the conductivities over the first 6 cycles. Answer: a6 = 729/64, S6 = 3367/64 Solution: Identify the first term a1 = 64 and the common ratio r = 3/4. The number of terms n = 6. Use the nth term formula a_n = a1 * r^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 64 and the common ratio r = 3/4. The number of terms n = 6. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=6, a6 = 64 * (3/4)^(6-1) = 64 * (3/4)^5. Step 3: Calculate (3/4)^5 = 3^5 / 4^5 = 243 / 1024. Step 4: Multiply by 64: a6 = 64 * 243 / 1024 = (64/1024) * 243 = (1/16) * 243 = 243/16. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=6, S6 = 64 * (1 - (3/4)^6) / (1 - 3/4). Step 6: Calculate (3/4)^6 = (3/4) * (3/4)^5 = (3/4) * 243/1024 = 729/4096. Step 7: Numerator: 1 - 729/4096 = (4096 - 729)/4096 = 3367/4096. Step 8: Denominator: 1 - 3/4 = 1/4. Step 9: (1 - r^n)/(1 - r) = (3367/4096) / (1/4) = (3367/4096) * 4 = 3367/1024. Step 10: Multiply by a1: S6 = 64 * 3367/1024 = (64/1024) * 3367 = (1/16) * 3367 = 3367/16. The conductivity after the 6th cycle is 243/16 S/m, and the sum of conductivities over the first 6 cycles is 3367/16 S/m.

4. Matiu is a marine biologist studying the population growth of a rare species of coral on a protected reef. He notes that the coral covers an area that grows geometrically each year. In the first year, the coral covers 24 square metres. Each subsequent year, the area covered increases by a factor of 2.5 times the previous year's area. Matiu needs to report the area covered after 10 years (the 10th term of the sequence) and the total area covered over the first 10 years for a conservation grant application. Find the area covered in the 10th year and the total area covered over the first 10 years. Answer: a10 = 366210.9375, S10 = 610351.5625 Solution: Identify the first term a1 = 24 and the common ratio r = 2.5. 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 = 24 and the common ratio r = 2.5. The number of terms n = 10. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=10, a10 = 24 * (2.5)^(10-1) = 24 * (2.5)^9. Step 3: Calculate (2.5)^9. 2.5 = 5/2. (5/2)^9 = 5^9 / 2^9 = 1953125 / 512 = 3814.697265625. Step 4: Multiply by 24: a10 = 24 * 3814.697265625 = 91552.734375. (Wait, recalculate carefully: 1953125/512 = 3814.697265625, then 24 * 3814.697265625 = 91552.734375. But check: 24 * 1953125 = 46875000, divided by 512 = 46875000/512 = 91552.734375. However, this seems too large. Let's recalculate (2.5)^9 step by step: 2.5^2 = 6.25, 2.5^4 = 6.25^2 = 39.0625, 2.5^8 = 39.0625^2 = 1525.87890625, then 2.5^9 = 1525.87890625 * 2.5 = 3814.697265625. So a10 = 24 * 3814.697265625 = 91552.734375. That is correct.) Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=10, S10 = 24 * (1 - 2.5^10) / (1 - 2.5). Step 6: Calculate 2.5^10 = 2.5 * 2.5^9 = 2.5 * 3814.697265625 = 9536.7431640625. Step 7: Numerator: 1 - 9536.7431640625 = -9535.7431640625. Denominator: 1 - 2.5 = -1.5. Step 8: (1 - r^n)/(1 - r) = (-9535.7431640625)/(-1.5) = 6357.162109375. Step 9: Multiply by a1: S10 = 24 * 6357.162109375 = 152571.890625. Step 10: Verify using alternate formula S_n = a1*(r^n - 1)/(r - 1): S10 = 24 * (9536.7431640625 - 1)/(2.5 - 1) = 24 * 9535.7431640625/1.5 = 24 * 6357.162109375 = 152571.890625. The area in the 10th year is 91552.734375 square metres and the total area over 10 years is 152571.890625 square metres.

5. A geometric sequence has its first term as 4 and a common ratio of 3. The sum of the first n terms of this sequence is 4372. Find the value of n. Answer: 7 Solution: First term a = 4 Common ratio r = 3 Sum of first n terms S_n = 4372 Recall the formula for the sum of the first n terms of a geometric sequence. S_n = a * (r^n - 1) / (r - 1) for r ≠ 1.
   Full step-by-step solution

   We are given: First term a = 4 Common ratio r = 3 Sum of first n terms S_n = 4372 --- **Step 1: Recall the formula for the sum of the first n terms of a geometric sequence.** The formula is: S_n = a * (r^n - 1) / (r - 1) for r ≠ 1. --- **Step 2: Substitute the known values into the formula.** 4372 = 4 * (3^n - 1) / (3 - 1) 4372 = 4 * (3^n - 1) / 2 --- **Step 3: Simplify the equation.** 4372 = 2 * (3^n - 1) Divide both sides by 2: 2186 = 3^n - 1 --- **Step 4: Solve for 3^n.** 2186 + 1 = 3^n 2187 = 3^n --- **Step 5: Recognize 2187 as a power of 3.** Check powers of 3: 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729 3^7 = 2187 So 2187 = 3^7. --- **Step 6: Conclude n.** 3^n = 3^7 n = 7 --- **Final answer:** n = 7