Arithmetic Sequences

lessonbunny.com

Answer Key & Explanations

Arithmetic Sequences · Grade 12 · Worksheet 2

1. 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

2. Emma's arithmetic sequence: a₁ = 15, d = -5. Write the explicit formula aₙ = ? Answer: aₙ = 20 - 5n Solution: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n - 1)d Substitute the given values: a₁ = 15 and d = -5 aₙ = 15 + (n - 1)(-5) Distribute the -5: aₙ = 15 - 5n + 5 Combine like terms: aₙ = 20 - 5n The explicit formula is aₙ = 20 - 5n.
   Full step-by-step solution

   Step 1: Recall the explicit formula for an arithmetic sequence: 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: Distribute the -5: aₙ = 15 - 5n + 5 Step 5: Combine like terms: aₙ = 20 - 5n The explicit formula is aₙ = 20 - 5n.

3. During a training program, Alex runs 287 meters on the first day and increases the distance by a constant amount each day. On day 15, Alex runs 4949 meters. What is the daily increase in distance? Answer: 333 Solution: This is an arithmetic sequence with first term a₁ = 287 and we need to find the common difference d. The explicit formula for an arithmetic sequence is a_n = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: This is an arithmetic sequence with first term a₁ = 287 and we need to find the common difference d. Step 2: The explicit formula for an arithmetic sequence is a_n = a₁ + (n-1)d. Step 3: Plug in the values: 4949 = 287 + (15 - 1)d Step 4: Subtract 287 from both sides: 4949 - 287 = (15 - 1)d Step 5: Divide by (15 - 1): d = (4949 - 287) / (15 - 1) Step 6: Calculate: d = (4949 - 287) // (15 - 1) = 333 The daily increase is 333 meters.

4. During a training program, Mason runs 293 meters on the first day and increases the distance by a constant amount each day. On day 8, Mason runs 3884 meters. What is the daily increase in distance? Answer: 513 Solution: This is an arithmetic sequence with first term a₁ = 293 and we need to find the common difference d. The explicit formula for an arithmetic sequence is a_n = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: This is an arithmetic sequence with first term a₁ = 293 and we need to find the common difference d. Step 2: The explicit formula for an arithmetic sequence is a_n = a₁ + (n-1)d. Step 3: Plug in the values: 3884 = 293 + (8 - 1)d Step 4: Subtract 293 from both sides: 3884 - 293 = (8 - 1)d Step 5: Divide by (8 - 1): d = (3884 - 293) / (8 - 1) Step 6: Calculate: d = (3884 - 293) // (8 - 1) = 513 The daily increase is 513 meters.

5. Given the arithmetic sequence: 15, 25, 35, 45, ... Find the 12th term. Answer: 125 Solution: Identify the common difference (d). 25 - 15 = 10 35 - 25 = 10 45 - 35 = 10 So, d = 10. Write the explicit formula for an arithmetic sequence.
   Full step-by-step solution

   Step 1: Identify the common difference (d). 25 - 15 = 10 35 - 25 = 10 45 - 35 = 10 So, d = 10. Step 2: Write the explicit formula for an arithmetic sequence. a_n = a_1 + (n - 1)d Step 3: Substitute the known values. a_1 = 15, d = 10, n = 12 a_12 = 15 + (12 - 1) * 10 Step 4: Calculate the value. a_12 = 15 + (11) * 10 a_12 = 15 + 110 a_12 = 125 The 12th term is 125.

6. A geometric pattern is formed by stacking spheres in a pyramid arrangement. The bottom layer is a square array of 8 spheres by 8 spheres. Each subsequent layer has one fewer sphere along each dimension than the layer below it. How many total spheres are in the pyramid when it reaches the top layer with a single sphere? Answer: 204 Solution: Identify the pattern. The bottom layer has 8 × 8 = 64 spheres. The next layer has 7 × 7 = 49 spheres, then 6 × 6 = 36 spheres, and so on, down to 1 × 1 = 1 sphere.
   Full step-by-step solution

   Step 1: Identify the pattern. The bottom layer has 8 × 8 = 64 spheres. The next layer has 7 × 7 = 49 spheres, then 6 × 6 = 36 spheres, and so on, down to 1 × 1 = 1 sphere. Step 2: Write out the sequence: 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² Step 3: Use the formula for the sum of squares: n(n+1)(2n+1)/6, where n = 8 Step 4: Substitute n = 8 into the formula: 8 × 9 × 17 / 6 Step 5: Calculate step by step: 8 × 9 = 72, 72 × 17 = 1224, 1224 / 6 = 204 Step 6: The total number of spheres is 204.

7. Arithmetic sequence: a₁=9, d=7. Write explicit formula aₙ = ? Answer: aₙ = 7n + 2 Solution: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 9 and d = 7 aₙ = 9 + (n-1)×7 Distribute the 7: aₙ = 9 + 7n - 7 Combine like terms: aₙ = 7n + 2 The explicit formula is aₙ = 7n + 2
   Full step-by-step solution

   Step 1: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Step 2: Substitute the given values: a₁ = 9 and d = 7 Step 3: aₙ = 9 + (n-1)×7 Step 4: Distribute the 7: aₙ = 9 + 7n - 7 Step 5: Combine like terms: aₙ = 7n + 2 Step 6: The explicit formula is aₙ = 7n + 2

8. Sophia's arithmetic sequence: a₁=11, d=6. Write the explicit formula for aₙ. Answer: aₙ = 6n + 5 Solution: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 11, d = 6 aₙ = 11 + (n-1)6 Distribute the 6: aₙ = 11 + 6n - 6 Combine like terms: aₙ = 6n + 5 The explicit formula is aₙ = 6n + 5.
   Full step-by-step solution

   Step 1: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Step 2: Substitute the given values: a₁ = 11, d = 6 Step 3: aₙ = 11 + (n-1)6 Step 4: Distribute the 6: aₙ = 11 + 6n - 6 Step 5: Combine like terms: aₙ = 6n + 5 The explicit formula is aₙ = 6n + 5.