Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 2

1. Arithmetic sequence: a₁=15, d=5. Write explicit formula aₙ = ? Answer: aₙ = 15 + (n-1)5 Solution: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 15 and d = 5 aₙ = 15 + (n-1)5 The formula can also be written as aₙ = 5n + 10 Both forms are correct, but the standard explicit form is aₙ = 15 + (n-1)5
   Full step-by-step solution

   Step 1: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d Step 2: Substitute the given values: a₁ = 15 and d = 5 Step 3: aₙ = 15 + (n-1)5 Step 4: The formula can also be written as aₙ = 5n + 10 Step 5: Both forms are correct, but the standard explicit form is aₙ = 15 + (n-1)5

2. 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 follows the function C(t) = 50te^(-0.2t), where t is time in hours after administration. The company needs to determine the maximum concentration reached and the time at which it occurs. Find both the time of maximum concentration and the maximum concentration value. Answer: t = 5 hours, C = 250/e mg/L ≈ 91.97 mg/L Solution: C(t) = 50 t e^(-0.2 t) We want the maximum concentration and the time t when it occurs.
   Full step-by-step solution

   Let's solve step by step. We are given: C(t) = 50 t e^(-0.2 t) We want the maximum concentration and the time t when it occurs. --- **Step 1: Find the derivative of C(t)** We use the product rule: If C(t) = u(t) v(t), where u(t) = 50 t v(t) = e^(-0.2 t) Then u'(t) = 50 v'(t) = -0.2 e^(-0.2 t) Product rule: C'(t) = u'(t) v(t) + u(t) v'(t) C'(t) = 50 e^(-0.2 t) + 50 t (-0.2 e^(-0.2 t)) C'(t) = 50 e^(-0.2 t) - 10 t e^(-0.2 t) Factor: C'(t) = e^(-0.2 t) (50 - 10 t) --- **Step 2: Set derivative equal to 0 to find critical points** e^(-0.2 t) is never 0, so: 50 - 10 t = 0 10 t = 50 t = 5 hours --- **Step 3: Verify it's a maximum** We can check the sign of C'(t): For t < 5, say t = 0: C'(0) = e^0 (50 - 0) = 50 > 0 (increasing) For t > 5, say t = 6: C'(6) = e^(-1.2) (50 - 60) = negative (decreasing) So t = 5 is a maximum. --- **Step 4: Find maximum concentration value** C(5) = 50 * 5 * e^(-0.2 * 5) C(5) = 250 * e^(-1) C(5) = 250 / e mg/L Numerical approximation: e ≈ 2.71828 250 / 2.71828 ≈ 91.97 mg/L --- **Final Answer:** Time of maximum concentration: t = 5 hours Maximum concentration: C = 250/e mg/L ≈ 91.97 mg/L

3. Mere's arithmetic sequence: 23, 37, 51, 65... Write the explicit formula aₙ = ? Answer: aₙ = 23 + (n-1)×14 Solution: Find the common difference d by subtracting consecutive terms: 37 - 23 = 14, 51 - 37 = 14, 65 - 51 = 14. So d = 14. Identify the first term a₁ = 23.
   Full step-by-step solution

   Step 1: Find the common difference d by subtracting consecutive terms: 37 - 23 = 14, 51 - 37 = 14, 65 - 51 = 14. So d = 14. Step 2: Identify the first term a₁ = 23. Step 3: Use the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d. Step 4: Substitute a₁ = 23 and d = 14: aₙ = 23 + (n-1)×14. Step 5: The explicit formula is aₙ = 23 + (n-1)×14.

4. Arithmetic sequence: a₁=14, d=9. Write explicit formula aₙ = ? Answer: aₙ = 14 + (n-1)9 Solution: Identify the given values: first term a₁ = 14, common difference d = 9 Use the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: aₙ = 14 + (n-1)9 The explicit formula is aₙ = 14 + (n-1)9
   Full step-by-step solution

   Step 1: Identify the given values: first term a₁ = 14, common difference d = 9 Step 2: Use the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Step 3: Substitute the given values: aₙ = 14 + (n-1)9 Step 4: The explicit formula is aₙ = 14 + (n-1)9

5. Aroha's arithmetic sequence: 7, 13, 19, 25... Find the 15th term. Answer: 91 Solution: The sequence is: 7, 13, 19, 25 13 - 7 = 6 19 - 13 = 6 25 - 19 = 6 The common difference d = 6 Identify the first term (a_1) The first term is 7, so a_1 = 7 a_n = a_1 + (n-1)d For the 15th term: a_15 = 7 + (15-1)×6 Calculate the 15th term a_15 = 7 + (14)×6 a_15 = 7 + 84 a_15 = 91 The 15th term of…
   Full step-by-step solution

   Step 1: Identify the common difference (d) The sequence is: 7, 13, 19, 25 13 - 7 = 6 19 - 13 = 6 25 - 19 = 6 The common difference d = 6 Step 2: Identify the first term (a_1) The first term is 7, so a_1 = 7 Step 3: Use the explicit formula for arithmetic sequences a_n = a_1 + (n-1)d For the 15th term: a_15 = 7 + (15-1)×6 Step 4: Calculate the 15th term a_15 = 7 + (14)×6 a_15 = 7 + 84 a_15 = 91 The 15th term of the sequence is 91.

6. Is 8, 15, 22, 29... arithmetic? If so, find the common difference d and write the explicit formula aₙ = ? Answer: Yes, d = 7, aₙ = 8 + (n-1)×7 Solution: Check if the sequence is arithmetic by finding the differences between consecutive terms: 15 - 8 = 7, 22 - 15 = 7, 29 - 22 = 7. The differences are all equal to 7, so the sequence is arithmetic.
   Full step-by-step solution

   Step 1: Check if the sequence is arithmetic by finding the differences between consecutive terms: 15 - 8 = 7, 22 - 15 = 7, 29 - 22 = 7. The differences are all equal to 7, so the sequence is arithmetic. Step 2: The common difference d = 7. Step 3: The first term a₁ = 8. Step 4: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d. Step 5: Substitute a₁ = 8 and d = 7: aₙ = 8 + (n-1)×7. The answer is: Yes, d = 7, aₙ = 8 + (n-1)×7.

7. 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) = 40te^(-0.1t), where t is time in hours after administration. The company needs to determine the time when the concentration reaches its maximum value. At what time does this maximum concentration occur? Answer: 10 Solution: Find the derivative of C(t) = 40te^(-0.1t) using the product rule. Let u = 40t and v = e^(-0.1t), then u' = 40 and v' = -0.1e^(-0.1t).
   Full step-by-step solution

   Step 1: Find the derivative of C(t) = 40te^(-0.1t) using the product rule. Step 2: Let u = 40t and v = e^(-0.1t), then u' = 40 and v' = -0.1e^(-0.1t). Step 3: Apply product rule: C'(t) = u'v + uv' = 40e^(-0.1t) + 40t(-0.1e^(-0.1t)) = 40e^(-0.1t) - 4te^(-0.1t). Step 4: Factor out 4e^(-0.1t): C'(t) = 4e^(-0.1t)(10 - t). Step 5: Set C'(t) = 0: 4e^(-0.1t)(10 - t) = 0. Step 6: Since 4e^(-0.1t) is never zero, we solve 10 - t = 0, giving t = 10. Step 7: Verify this is a maximum by checking the sign of C'(t) around t = 10. The maximum concentration occurs at t = 10 hours.

8. Aroha is creating a visual art piece using a triangular arrangement of identical circular tiles. The topmost row has 1 tile, the second row has 3 tiles, the third row has 5 tiles, and the fourth row has 7 tiles. If the pattern continues in this arithmetic manner and there are 9 rows in total, how many tiles does Aroha need to complete the entire piece? Answer: 81 Solution: Identify the sequence of tiles per row: 1, 3, 5, 7, ... This is an arithmetic sequence with first term a1 = 1 and common difference d = 2. The explicit formula for the nth row is a_n = a1 + (n-1)d = 1 + (n-1)*2 = 2n - 1.
   Full step-by-step solution

   Step 1: Identify the sequence of tiles per row: 1, 3, 5, 7, ... Step 2: This is an arithmetic sequence with first term a1 = 1 and common difference d = 2. Step 3: The explicit formula for the nth row is a_n = a1 + (n-1)d = 1 + (n-1)*2 = 2n - 1. Step 4: For 9 rows, the last term (row 9) is a9 = 2*9 - 1 = 17. Step 5: The sum of the first 9 terms of an arithmetic series is S_n = n/2 * (first term + last term). Step 6: Substitute n = 9, first term = 1, last term = 17: S9 = 9/2 * (1 + 17) = 9/2 * 18. Step 7: Calculate: 9/2 * 18 = 9 * 9 = 81. Step 8: Aroha needs 81 tiles in total. The answer is 81.