Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 3

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

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

2. Matiu is saving up to buy a new video game that costs $463. Matiu starts with $71 and adds the same amount of money to the savings account every week. After 28 weeks, Matiu has exactly enough money to buy the game. How much money does Matiu add each week? Answer: 14 Solution: Recognize this is an arithmetic sequence where the total saved after 28 weeks is the sum of the first 28 terms. The amount added each week is the common difference d.
   Full step-by-step solution

   Step 1: Recognize this is an arithmetic sequence where the total saved after 28 weeks is the sum of the first 28 terms. Step 2: The amount added each week is the common difference d. Step 3: The total saved after 28 weeks: S_n = (n/2) * (2*71 + (n-1)d) = 463. Step 4: Plug in: (28/2) * (2*71 + (28-1)d) = 463. Step 5: Multiply both sides by 2: 28 * (2*71 + (28-1)d) = 2*463. Step 6: Divide by 28: 2*71 + (28-1)d = (2*463) // 28. Step 7: Subtract 2*71: (28-1)d = (2*463) // 28 - 2*71. Step 8: Divide by (28-1): d = [(2*463) // 28 - 2*71] // (28-1). Step 9: Alternatively, since total = start + n*d, we get d = (463 - 71) // 28. Step 10: Therefore, the weekly addition is $14.

3. Aroha's arithmetic sequence: 9, 15, 21, 27... Write the explicit formula aₙ = ? Answer: aₙ = 9 + (n-1)×6 Solution: Identify the first term a₁ = 9. Find the common difference d by subtracting consecutive terms: 15 - 9 = 6, 21 - 15 = 6, 27 - 21 = 6. So d = 6.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 9. Step 2: Find the common difference d by subtracting consecutive terms: 15 - 9 = 6, 21 - 15 = 6, 27 - 21 = 6. So d = 6. Step 3: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d. Step 4: Substitute a₁ = 9 and d = 6: aₙ = 9 + (n-1)×6. Step 5: This can be simplified to aₙ = 6n + 3, but the standard explicit form is aₙ = 9 + (n-1)×6. The answer is aₙ = 9 + (n-1)×6.

4. Liam is analyzing the growth of a bacterial culture in a biology lab. The population P(t) (in thousands) at time t hours is modeled by the function P(t) = 5e^(0.2t). Using calculus, determine the instantaneous rate of change of the population when t = 3 hours. Express your answer in thousands of bacteria per hour. Answer: 1.822 Solution: P(t) = 5 * e^(0.2t) where P(t) is in thousands of bacteria, and t is in hours. The instantaneous rate of change of P(t) at a specific time t is the derivative P'(t).
   Full step-by-step solution

   We are given the population function: P(t) = 5 * e^(0.2t) where P(t) is in thousands of bacteria, and t is in hours. The instantaneous rate of change of P(t) at a specific time t is the derivative P'(t). --- **Step 1: Differentiate P(t)** The derivative of e^(0.2t) with respect to t is 0.2 * e^(0.2t) by the chain rule. Since 5 is a constant multiple: P'(t) = 5 * 0.2 * e^(0.2t) P'(t) = 1 * e^(0.2t) So P'(t) = e^(0.2t) --- **Step 2: Evaluate P'(t) at t = 3** P'(3) = e^(0.2 * 3) P'(3) = e^(0.6) --- **Step 3: Compute numerical value** We know e^0.6 is approximately 1.8221188 (using a calculator or known value). Rounding to three decimal places: 1.822 --- **Step 4: Interpret units** P'(t) is in thousands of bacteria per hour, so the instantaneous rate of change at t = 3 hours is 1.822 thousands of bacteria per hour. --- **Final answer:** 1.822

5. Moana is stacking cans in a grocery store display. The top row has 13 cans, and each row below has 4 more cans than the row above it. If the bottom row has 37 cans, how many rows are in the display? Answer: 7 Solution: This is an arithmetic sequence where the number of cans per row follows a_n = a₁ + (n-1)d. Here, a₁ = 13 (top row), a_n = 37 (bottom row), and d = 4. Use the formula: a_n = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: This is an arithmetic sequence where the number of cans per row follows a_n = a₁ + (n-1)d. Step 2: Here, a₁ = 13 (top row), a_n = 37 (bottom row), and d = 4. Step 3: Use the formula: a_n = a₁ + (n-1)d. Step 4: Substitute: 37 = 13 + (n-1)*4. Step 5: Subtract 13 from both sides: 37 - 13 = (n-1)*4. Step 6: Divide by 4: (37 - 13) // 4 = n - 1. Step 7: Add 1: n = (37 - 13) // 4 + 1. Step 8: Calculate: n = 7. Step 9: Therefore, the display has 7 rows.

6. Sarah is stacking cans in a grocery store display. The top row has 7 cans, and each row below has 6 more cans than the row above it. If the bottom row has 37 cans, how many rows are in the display? Answer: 6 Solution: This is an arithmetic sequence where the number of cans per row follows a_n = a₁ + (n-1)d. Here, a₁ = 7 (top row), a_n = 37 (bottom row), and d = 6. Use the formula: a_n = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: This is an arithmetic sequence where the number of cans per row follows a_n = a₁ + (n-1)d. Step 2: Here, a₁ = 7 (top row), a_n = 37 (bottom row), and d = 6. Step 3: Use the formula: a_n = a₁ + (n-1)d. Step 4: Substitute: 37 = 7 + (n-1)*6. Step 5: Subtract 7 from both sides: 37 - 7 = (n-1)*6. Step 6: Divide by 6: (37 - 7) // 6 = n - 1. Step 7: Add 1: n = (37 - 7) // 6 + 1. Step 8: Calculate: n = 6. Step 9: Therefore, the display has 6 rows.

7. During a training program, Mere runs 483 meters on the first day and increases the distance by a constant amount each day. On day 11, Mere runs 1433 meters. What is the daily increase in distance? Answer: 95 Solution: This is an arithmetic sequence with first term a₁ = 483 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 = 483 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: 1433 = 483 + (11 - 1)d Step 4: Subtract 483 from both sides: 1433 - 483 = (11 - 1)d Step 5: Divide by (11 - 1): d = (1433 - 483) / (11 - 1) Step 6: Calculate: d = (1433 - 483) // (11 - 1) = 95 The daily increase is 95 meters.

8. Given the arithmetic sequence: 7, 15, 23, 31, ... Write the explicit formula aₙ. Answer: aₙ = 8n - 1 Solution: Identify the first term: a₁ = 7 Find the common difference: d = 15 - 7 = 8 Apply the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the values: aₙ = 7 + (n-1)8 Simplify: aₙ = 7 + 8n - 8 Combine like terms: aₙ = 8n - 1 The explicit formula is aₙ = 8n - 1.
   Full step-by-step solution

   Step 1: Identify the first term: a₁ = 7 Step 2: Find the common difference: d = 15 - 7 = 8 Step 3: Apply the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Step 4: Substitute the values: aₙ = 7 + (n-1)8 Step 5: Simplify: aₙ = 7 + 8n - 8 Step 6: Combine like terms: aₙ = 8n - 1 The explicit formula is aₙ = 8n - 1.

9. Arithmetic sequence: a₁=8, d=5. Write explicit formula aₙ = ? Answer: aₙ = 5n + 3 Solution: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 8 and d = 5 aₙ = 8 + (n-1)5 Distribute the 5: aₙ = 8 + 5n - 5 Combine like terms: aₙ = 5n + 3 The explicit formula is aₙ = 5n + 3
   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 = 8 and d = 5 Step 3: aₙ = 8 + (n-1)5 Step 4: Distribute the 5: aₙ = 8 + 5n - 5 Step 5: Combine like terms: aₙ = 5n + 3 The explicit formula is aₙ = 5n + 3