Geometric Sequences

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Answer Key & Explanations

Geometric Sequences · Grade 12 · Worksheet 3

1. ∑(n=1 to 6) 8 × (-2)^(n-1) = ? Answer: -168 Solution: a₁ = 8 × (-2)^(1-1) = 8 × (-2)^0 = 8 × 1 = 8 r = -2 Sₙ = a₁ × (1 - rⁿ) / (1 - r) Substitute n = 6, a₁ = 8, r = -2 S₆ = 8 × (1 - (-2)⁶) / (1 - (-2)) Calculate (-2)⁶ (-2)⁶ = 64 S₆ = 8 × (1 - 64) / (1 + 2) S₆ = 8 × (-63) / 3 S₆ = -504 / 3 S₆ = -168 The answer is -168.
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and common ratio (r) a₁ = 8 × (-2)^(1-1) = 8 × (-2)^0 = 8 × 1 = 8 r = -2 Step 2: Use the geometric series sum formula for n terms Sₙ = a₁ × (1 - rⁿ) / (1 - r) Step 3: Substitute n = 6, a₁ = 8, r = -2 S₆ = 8 × (1 - (-2)⁶) / (1 - (-2)) Step 4: Calculate (-2)⁶ (-2)⁶ = 64 Step 5: Substitute into the formula S₆ = 8 × (1 - 64) / (1 + 2) S₆ = 8 × (-63) / 3 S₆ = -504 / 3 S₆ = -168 The answer is -168.

2. ∑(n=1 to 6) 5 × (-2)^(n-1) = ? Answer: -105 Solution: a₁ = 5 × (-2)^(1-1) = 5 × (-2)^0 = 5 × 1 = 5 r = -2 Sₙ = a₁ × (1 - rⁿ) / (1 - r) Substitute n = 6, a₁ = 5, r = -2 S₆ = 5 × (1 - (-2)⁶) / (1 - (-2)) Calculate (-2)⁶ (-2)⁶ = 64 S₆ = 5 × (1 - 64) / (1 + 2) S₆ = 5 × (-63) / 3 S₆ = 5 × (-21) S₆ = -105 Term 1: 5 × (-2)^0 = 5 Term 2: 5 × (-2)^1 = -10…
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and common ratio (r) a₁ = 5 × (-2)^(1-1) = 5 × (-2)^0 = 5 × 1 = 5 r = -2 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 = -2 S₆ = 5 × (1 - (-2)⁶) / (1 - (-2)) Step 4: Calculate (-2)⁶ (-2)⁶ = 64 Step 5: Substitute into the formula S₆ = 5 × (1 - 64) / (1 + 2) S₆ = 5 × (-63) / 3 S₆ = 5 × (-21) S₆ = -105 Step 6: Verify by calculating individual terms Term 1: 5 × (-2)^0 = 5 Term 2: 5 × (-2)^1 = -10 Term 3: 5 × (-2)^2 = 20 Term 4: 5 × (-2)^3 = -40 Term 5: 5 × (-2)^4 = 80 Term 6: 5 × (-2)^5 = -160 Sum = 5 + (-10) + 20 + (-40) + 80 + (-160) = -5 + 20 + (-40) + 80 + (-160) = 15 + (-40) + 80 + (-160) = -25 + 80 + (-160) = 55 + (-160) = -105 The answer is -105.

3. Mason is a materials engineer analyzing the growth of a bacterial biofilm on a metallic surface. He observes that the biofilm area doubles every day. On the first day of measurement, the biofilm covers 7 square millimeters. If this geometric growth pattern continues, what is the total area covered by the biofilm over the first 7 days (including day 1)? Answer: 889 Solution: Identify the first term a1 = 7 and the common ratio r = 2. The number of terms n = 7.
   Full step-by-step solution

   Step 1: Identify the first term a1 = 7 and the common ratio r = 2. The number of terms n = 7. Step 2: Use the geometric series sum formula: Sn = a1 * (r^n - 1) / (r - 1) Step 3: Substitute the values: S7 = 7 * (2^7 - 1) / (2 - 1) Step 4: Calculate 2^7 = 2 * 2 * 2 * 2 * 2 * 2 * 2 = 128 Step 5: Compute the numerator: 128 - 1 = 127 Step 6: Compute the denominator: 2 - 1 = 1 Step 7: Complete the calculation: S7 = 7 * (127 / 1) = 7 * 127 = 889 The total area covered over the first 7 days is 889 square millimeters.

4. ∑(n=1 to 5) 4 × (1/2)^(n-1) = ? Answer: 7.75 Solution: Identify the first term a₁ = 4 and common ratio r = 1/2 Use the finite geometric series formula S_n = a₁(1 - rⁿ)/(1 - r) Substitute n = 5: S₅ = 4(1 - (1/2)⁵)/(1 - 1/2) Calculate (1/2)⁵ = 1/32 = 0.03125 Calculate 1 - 0.03125 = 0.96875 Calculate 1 - 1/2 = 1/2 = 0.5 S₅ = 4 × 0.96875 / 0.5 4 ×…
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 4 and common ratio r = 1/2 Step 2: Use the finite geometric series formula S_n = a₁(1 - rⁿ)/(1 - r) Step 3: Substitute n = 5: S₅ = 4(1 - (1/2)⁵)/(1 - 1/2) Step 4: Calculate (1/2)⁵ = 1/32 = 0.03125 Step 5: Calculate 1 - 0.03125 = 0.96875 Step 6: Calculate 1 - 1/2 = 1/2 = 0.5 Step 7: S₅ = 4 × 0.96875 / 0.5 Step 8: 4 × 0.96875 = 3.875 Step 9: 3.875 ÷ 0.5 = 7.75 The answer is 7.75.

5. Aroha is a marine biologist monitoring the population of a rare species of sea star in a protected marine reserve. She notes that the number of sea stars follows a geometric growth pattern. In the first year of her study, she counts 9 sea stars. Each subsequent year, the population increases by a factor of 4 times the previous year's count. Aroha needs to report the population in the 10th year (the 10th term of the sequence) and the total number of sea stars observed 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. 4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024, 4^6 = 4096, 4^7 = 16384, 4^8 = 65536, 4^9 = 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: Verify using alternate formula: S10 = a1 * (r^n - 1)/(r - 1) = 9 * (1048576 - 1)/(4 - 1) = 9 * 1048575/3 = 9 * 349525 = 3145725. The population in the 10th year is 2359296 sea stars and the total population over the first 10 years is 3145725 sea stars.

6. Mere is a marine biologist studying the population growth of a rare species of sea star in a protected marine reserve. She observes that the number of sea stars in a specific quadrat follows a geometric pattern. In the first year of her study, there are 8 sea stars. Each subsequent year, the population increases by a factor of 4 (the number of sea stars quadruples each year). Mere wants to know the population size after 6 years (the 6th term of the sequence) and the total number of sea stars that have been present in the quadrat over the first 6 years. Find the population in the 6th year and the total population over the first 6 years. Answer: a6 = 32768, S6 = 43688 Solution: Identify the first term a1 = 8 and the common ratio r = 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 = 8 and the common ratio r = 4. The number of terms n = 6. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=6, a6 = 8 * 4^(6-1) = 8 * 4^5. Step 3: Calculate 4^5 = 4 * 4 * 4 * 4 * 4 = 1024. Step 4: Multiply by 8: a6 = 8 * 1024 = 8192. Step 5: Use the sum formula S_n = a1 * (r^n - 1) / (r - 1). For n=6, S6 = 8 * (4^6 - 1) / (4 - 1). Step 6: Calculate 4^6 = 4 * 4^5 = 4 * 1024 = 4096. Step 7: Numerator: 4096 - 1 = 4095. Denominator: 4 - 1 = 3. Step 8: Divide: 4095 / 3 = 1365. Step 9: Multiply by a1: S6 = 8 * 1365 = 10920. The population in the 6th year is 8192 sea stars and the total population over the first 6 years is 10920 sea stars.