Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 1

1. During a training program, Ava runs 293 meters on the first day and increases the distance by a constant amount each day. On day 15, Ava runs 1077 meters. What is the daily increase in distance? Answer: 56 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: 1077 = 293 + (15 - 1)d Step 4: Subtract 293 from both sides: 1077 - 293 = (15 - 1)d Step 5: Divide by (15 - 1): d = (1077 - 293) / (15 - 1) Step 6: Calculate: d = (1077 - 293) // (15 - 1) = 56 The daily increase is 56 meters.

2. Liam is analyzing the growth of a bacterial culture in a laboratory experiment. The population P(t) (in thousands) after t hours is modeled by the function P(t) = 5e^(0.2t). At what instantaneous rate is the population growing when t = 10 hours? Express your answer in thousands of bacteria per hour. Answer: 10e^2 Solution: The instantaneous rate of change of a function at a given point is found using differentiation. For exponential functions of the form f(x) = ae^(bx), the derivative f'(x) = abe^(bx) gives the rate of change at any point x.
   Full step-by-step solution

   The instantaneous rate of change of a function at a given point is found using differentiation. For exponential functions of the form f(x) = ae^(bx), the derivative f'(x) = abe^(bx) gives the rate of change at any point x. This concept is fundamental in modeling real-world phenomena like population growth, where we need to know how quickly quantities are changing at precise moments.

3. 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

4. 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.

5. Olivia is saving up to buy a new video game that costs $376. Olivia starts with $70 and adds the same amount of money to the savings account every week. After 51 weeks, Olivia has exactly enough money to buy the game. How much money does Olivia add each week? Answer: 6 Solution: Recognize this is an arithmetic sequence where the total saved after 51 weeks is the sum of the first 51 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 51 weeks is the sum of the first 51 terms. Step 2: The amount added each week is the common difference d. Step 3: The total saved after 51 weeks: S_n = (n/2) * (2*70 + (n-1)d) = 376. Step 4: Plug in: (51/2) * (2*70 + (51-1)d) = 376. Step 5: Multiply both sides by 2: 51 * (2*70 + (51-1)d) = 2*376. Step 6: Divide by 51: 2*70 + (51-1)d = (2*376) // 51. Step 7: Subtract 2*70: (51-1)d = (2*376) // 51 - 2*70. Step 8: Divide by (51-1): d = [(2*376) // 51 - 2*70] // (51-1). Step 9: Alternatively, since total = start + n*d, we get d = (376 - 70) // 51. Step 10: Therefore, the weekly addition is $6.

6. A particle moves along a straight line such that its velocity at time t seconds is given by v(t) = 3t² - 12t + 9 m/s. The particle's displacement from its initial position after 4 seconds can be found by evaluating the definite integral of v(t) from 0 to 4. Calculate this displacement. Answer: 4 Solution: The velocity function is v(t) = 3t² - 12t + 9. Displacement from t = 0 to t = 4 is the definite integral of v(t) from 0 to 4. Displacement = ∫ from 0 to 4 of (3t² - 12t + 9) dt.
   Full step-by-step solution

   Step 1: Understand the problem The velocity function is v(t) = 3t² - 12t + 9. Displacement from t = 0 to t = 4 is the definite integral of v(t) from 0 to 4. Step 2: Set up the integral Displacement = ∫ from 0 to 4 of (3t² - 12t + 9) dt. Step 3: Find the antiderivative The antiderivative F(t) of 3t² is 3*(t³/3) = t³. The antiderivative of -12t is -12*(t²/2) = -6t². The antiderivative of 9 is 9t. So F(t) = t³ - 6t² + 9t. Step 4: Evaluate the definite integral Displacement = F(4) - F(0). First compute F(4): F(4) = (4)³ - 6*(4)² + 9*(4) = 64 - 6*16 + 36 = 64 - 96 + 36 = (64 + 36) - 96 = 100 - 96 = 4. Now compute F(0): F(0) = (0)³ - 6*(0)² + 9*(0) = 0. Step 5: Subtract Displacement = F(4) - F(0) = 4 - 0 = 4. Final answer: The displacement after 4 seconds is 4 meters.

7. Arithmetic sequence: a₁=11, d=8. Write explicit formula Answer: aₙ = 11 + (n-1)8 Solution: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 11 and d = 8 The formula becomes: aₙ = 11 + (n-1)8 This is the explicit formula that can be used to find any term in the sequence The explicit formula is aₙ = 11 + (n-1)8
   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 and d = 8 Step 3: The formula becomes: aₙ = 11 + (n-1)8 Step 4: This is the explicit formula that can be used to find any term in the sequence The explicit formula is aₙ = 11 + (n-1)8

8. Aroha's arithmetic sequence: a₁ = 14, d = 9. Write the explicit formula aₙ = ? Answer: aₙ = 9n + 5 Solution: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n - 1)d Substitute the given values: a₁ = 14 and d = 9 aₙ = 14 + (n - 1) × 9 Distribute the 9: aₙ = 14 + 9n - 9 Combine like terms: aₙ = 9n + 5 The explicit formula is aₙ = 9n + 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 = 14 and d = 9 Step 3: aₙ = 14 + (n - 1) × 9 Step 4: Distribute the 9: aₙ = 14 + 9n - 9 Step 5: Combine like terms: aₙ = 9n + 5 The explicit formula is aₙ = 9n + 5.

9. Isabella's arithmetic sequence: a₁ = 11, d = 4. Write the explicit formula aₙ = ? Answer: aₙ = 4n + 7 Solution: Start with the general formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 11 and d = 4 aₙ = 11 + (n-1)(4) Distribute the 4: aₙ = 11 + 4n - 4 Combine like terms: aₙ = 4n + 7 The explicit formula is aₙ = 4n + 7.
   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 = 11 and d = 4 Step 3: aₙ = 11 + (n-1)(4) Step 4: Distribute the 4: aₙ = 11 + 4n - 4 Step 5: Combine like terms: aₙ = 4n + 7 The explicit formula is aₙ = 4n + 7.