Arithmetic Sequences

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Answer Key & Explanations

Arithmetic Sequences · Grade 12 · Worksheet 2

1. Sequence: 9, 18, 27, 36... Find a₁₇ and sum of first 17 terms. Answer: a₁₇ = 153, S₁₇ = 1377 Solution: Identify the first term a₁ = 9. Find the common difference d = 18 - 9 = 9. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 9. Step 2: Find the common difference d = 18 - 9 = 9. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 17: a₁₇ = 9 + (17-1)×9 = 9 + 16×9 = 9 + 144 = 153. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 17: S₁₇ = 17(9 + 153)/2 = 17×162/2 = 2754/2 = 1377. Therefore, a₁₇ = 153 and S₁₇ = 1377.

2. Sequence: 2, 8, 14, 20... Find a₁₈ and sum of first 18 terms. Answer: a₁₈ = 104, S₁₈ = 954 Solution: Identify the first term a₁ = 2. Find the common difference d = 8 - 2 = 6. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 2. Step 2: Find the common difference d = 8 - 2 = 6. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 18: a₁₈ = 2 + (18-1)×6 = 2 + 17×6 = 2 + 102 = 104. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 18: S₁₈ = 18(2 + 104)/2 = 18×106/2 = 1908/2 = 954. Therefore, a₁₈ = 104 and S₁₈ = 954.

3. A civil engineer is designing a suspension bridge where the main cable follows the curve y = 2x² - 8x + 10 meters above the water level, where x is the horizontal distance from the left tower in meters. To determine the minimum clearance between the cable and the water, the engineer needs to find the vertex of this parabola. What is the minimum height of the cable above the water? Answer: 2 Solution: Identify the coefficients from the quadratic function y = 2x² - 8x + 10 a = 2, b = -8, c = 10 Find the x-coordinate of the vertex using x = -b/(2a) x = -(-8)/(2×2) = 8/4 = 2 Substitute x = 2 into the original equation to find the minimum height y = 2(2)² - 8(2) + 10 y = 2(4) - 16 + 10 y = 8 - 16…
   Full step-by-step solution

   Step 1: Identify the coefficients from the quadratic function y = 2x² - 8x + 10 a = 2, b = -8, c = 10 Step 2: Find the x-coordinate of the vertex using x = -b/(2a) x = -(-8)/(2×2) = 8/4 = 2 Step 3: Substitute x = 2 into the original equation to find the minimum height y = 2(2)² - 8(2) + 10 y = 2(4) - 16 + 10 y = 8 - 16 + 10 y = 2 Step 4: The minimum height of the cable above the water is 2 meters. The answer is 2.

4. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is given by the function C(t) = 5te^(-0.2t), where t is time in hours after administration. The company needs to determine the maximum concentration of the drug in the bloodstream and when it occurs. Find the time t (in hours) when the concentration reaches its maximum value. Answer: 5 Solution: C(t) = 5 * t * e^(-0.2 * t) We want to find the time t when C(t) reaches its maximum. Since C(t) is a product of two functions of t, we use the product rule: Let f(t) = 5t, g(t) = e^(-0.2t).
   Full step-by-step solution

   We are given the concentration function: C(t) = 5 * t * e^(-0.2 * t) We want to find the time t when C(t) reaches its maximum. --- **Step 1: Find the derivative of C(t)** Since C(t) is a product of two functions of t, we use the product rule: Let f(t) = 5t, g(t) = e^(-0.2t). Then f'(t) = 5, and g'(t) = -0.2 * e^(-0.2t). Product rule: C'(t) = f'(t) * g(t) + f(t) * g'(t) C'(t) = 5 * e^(-0.2t) + 5t * (-0.2 * e^(-0.2t)) C'(t) = 5e^(-0.2t) - 1.0 * t * e^(-0.2t) C'(t) = e^(-0.2t) * (5 - t) --- **Step 2: Set derivative equal to 0 to find critical points** e^(-0.2t) is never 0, so: 5 - t = 0 t = 5 --- **Step 3: Verify that t = 5 gives a maximum** We can check the sign of C'(t) around t = 5: For t < 5, say t = 4: C'(4) = e^(-0.8) * (5 - 4) = positive For t > 5, say t = 6: C'(6) = e^(-1.2) * (5 - 6) = negative Since C'(t) changes from positive to negative at t = 5, it is a maximum. --- **Step 4: Conclusion** The maximum concentration occurs at t = 5 hours. **Final answer:** 5

5. Noah is saving money to buy a high-end telescope. He starts with $47 in his savings account and adds $12 every week. The amount in his account after n weeks forms an arithmetic sequence. How much money will Noah have after 30 weeks? What is the total amount he will have deposited over the 30 weeks? Answer: Noah will have $395 after 30 weeks, and the total amount deposited over 30 weeks is $6,630. Solution: Identify the first term a1 = 47 (initial amount) and common difference d = 12 (weekly addition). Step 2: Use the nth term formula an = a1 + (n-1)d. For n = 30: a30 = 47 + (30-1)*12 = 47 + 29*12 = 47 + 348 = 395.
   Full step-by-step solution

   Step 1: Identify the first term a1 = 47 (initial amount) and common difference d = 12 (weekly addition). Step 2: Use the nth term formula an = a1 + (n-1)d. For n = 30: a30 = 47 + (30-1)*12 = 47 + 29*12 = 47 + 348 = 395. So after 30 weeks, Noah has $395. Step 3: Use the sum formula Sn = n(a1 + an)/2 for the first 30 terms: S30 = 30(47 + 395)/2 = 30(442)/2 = 30*221 = 6630. The total amount deposited over 30 weeks is $6,630. Answer: Noah will have $395 after 30 weeks, and the total deposited is $6,630.

6. Charlotte is a financial planner who is helping a client save for a major purchase. The client plans to deposit money into a savings account each month, with the deposits forming an arithmetic sequence. The first deposit is $12, and each subsequent deposit increases by $7 from the previous month. Charlotte needs to determine the amount of the 17th deposit and the total amount deposited over the first 17 months to present a savings projection to the client. What is the amount of the 17th deposit, and what is the total amount deposited over the first 17 months? Answer: 124 and 1156 Solution: Identify the given values. First deposit a1 = 12, common difference d = 7, number of terms n = 17. Find the 17th deposit using the formula a_n = a1 + (n - 1)d.
   Full step-by-step solution

   Step 1: Identify the given values. First deposit a1 = 12, common difference d = 7, number of terms n = 17. Step 2: Find the 17th deposit using the formula a_n = a1 + (n - 1)d. a_17 = 12 + (17 - 1) * 7 a_17 = 12 + 16 * 7 a_17 = 12 + 112 a_17 = 124 Step 3: Find the sum of the first 17 deposits using the formula S_n = n(a1 + a_n)/2. S_17 = 17(12 + 124)/2 S_17 = 17(136)/2 S_17 = 2312/2 S_17 = 1156 Step 4: The 17th deposit is $124, and the total deposited over 17 months is $1156. The answer is 124 and 1156.

7. Sequence: 12, 17, 22, 27... Find a₁₂ and sum of first 12 terms. Answer: a₁₂ = 67, S₁₂ = 474 Solution: Identify the first term a₁ = 12. Find the common difference d = 17 - 12 = 5. Calculate the 12th term using aₙ = a₁ + (n-1)d: a₁₂ = 12 + (12-1)×5 = 12 + 11×5 = 12 + 55 = 67.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 12. Step 2: Find the common difference d = 17 - 12 = 5. Step 3: Calculate the 12th term using aₙ = a₁ + (n-1)d: a₁₂ = 12 + (12-1)×5 = 12 + 11×5 = 12 + 55 = 67. Step 4: Calculate the sum of the first 12 terms using Sₙ = n(a₁ + aₙ)/2: S₁₂ = 12(12 + 67)/2 = 12×79/2 = 948/2 = 474. Therefore, a₁₂ = 67 and S₁₂ = 474.