Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 3

1. A biomedical researcher is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = 80e^(-0.15t) + 20e^(-0.03t) mg/L, where t is time in hours. The researcher needs to calculate the total drug exposure over the first 24 hours, which is represented by the area under the concentration curve. Find the definite integral of C(t) from t=0 to t=24 to determine the total drug exposure. Answer: approximately 533.33 Solution: In pharmacokinetics, the area under the concentration-time curve represents total drug exposure. Exponential decay functions integrate using the formula ∫e^(kt)dt = (1/k)e^(kt) + C.
   Full step-by-step solution

   In pharmacokinetics, the area under the concentration-time curve represents total drug exposure. Exponential decay functions integrate using the formula ∫e^(kt)dt = (1/k)e^(kt) + C. When dealing with multiple exponential terms in a sum, each term can be integrated independently. The definite integral from a to b gives the net accumulation over that time interval, which in medical applications helps determine appropriate dosing regimens.

2. Mere is training for a marathon and records her weekly running distances. In week 1, she runs 12 km. Each subsequent week, she increases her distance by a constant amount. By week 16, she is running 87 km per week. What is the total distance Mere runs over the first 16 weeks? Answer: 792 Solution: Identify the first term and the 16th term. a1 = 12, a16 = 87, n = 16. Use the nth term formula: an = a1 + (n - 1)d.
   Full step-by-step solution

   Step 1: Identify the first term and the 16th term. a1 = 12, a16 = 87, n = 16. Step 2: Use the nth term formula: an = a1 + (n - 1)d. So 87 = 12 + (16 - 1)d. Step 3: Simplify: 87 = 12 + 15d. Subtract 12: 75 = 15d. Divide by 15: d = 5. Step 4: Now use the sum formula: Sn = n(a1 + an)/2. So S16 = 16(12 + 87)/2. Step 5: Calculate: 12 + 87 = 99. Then 16 * 99 = 1584. Divide by 2: 1584 / 2 = 792. The answer is 792.

3. Sequence: 3, 9, 15, 21... Find a₁₉ and sum of first 19 terms. Answer: a₁₉ = 111, S₁₉ = 1083 Solution: Identify the first term a₁ = 3. Find the common difference d = 9 - 3 = 6. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 3. Step 2: Find the common difference d = 9 - 3 = 6. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 19: a₁₉ = 3 + (19-1)×6 = 3 + 18×6 = 3 + 108 = 111. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 19: S₁₉ = 19(3 + 111)/2 = 19×114/2 = 2166/2 = 1083. Therefore, a₁₉ = 111 and S₁₉ = 1083.

4. A geometric pattern is formed by stacking triangular layers. The first layer has 1 equilateral triangle with side length 2 cm. The second layer has 3 equilateral triangles, each with side length 1 cm, arranged to form a larger triangle. The third layer has 5 equilateral triangles, each with side length 0.5 cm, continuing this pattern. If the pattern continues infinitely with each subsequent layer having 2 more triangles than the previous layer and each triangle having half the side length of the triangles in the previous layer, what is the total area of all triangles in this infinite series? (Area of equilateral triangle = (√3/4) × side²) Answer: 4√3 Solution: This problem involves analyzing an infinite geometric series where both the number of elements and their individual values follow geometric progressions. When dealing with infinite series of this type, we can use the formula for the sum of an infinite geometric series, provided the common ratio…
   Full step-by-step solution

   This problem involves analyzing an infinite geometric series where both the number of elements and their individual values follow geometric progressions. The key insight is recognizing that the total area in each layer forms its own geometric sequence. When dealing with infinite series of this type, we can use the formula for the sum of an infinite geometric series, provided the common ratio has an absolute value less than 1. The challenge is properly accounting for how both the count of triangles and their individual areas contribute to the total area progression.

5. Sequence: 9, 17, 25, 33... Find a₁₄ and sum of first 14 terms. Answer: a₁₄ = 113, S₁₄ = 854 Solution: Identify the first term a₁ = 9. Find the common difference d = 17 - 9 = 8. 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 = 17 - 9 = 8. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 14: a₁₄ = 9 + (14-1)×8 = 9 + 13×8 = 9 + 104 = 113. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 14: S₁₄ = 14(9 + 113)/2 = 14×122/2 = 1708/2 = 854. Therefore, a₁₄ = 113 and S₁₄ = 854.

6. Sequence: 10, 25, 40, 55... Find a₁₂ and sum of first 12 terms. Answer: a₁₂ = 175, S₁₂ = 1110 Solution: Identify the first term a₁ = 10. Find the common difference d = 25 - 10 = 15. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 10. Step 2: Find the common difference d = 25 - 10 = 15. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 12: a₁₂ = 10 + (12-1)×15 = 10 + 11×15 = 10 + 165 = 175. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 12: S₁₂ = 12(10 + 175)/2 = 12×185/2 = 2220/2 = 1110. Therefore, a₁₂ = 175 and S₁₂ = 1110.

7. Sequence: 3, 11, 19, 27... Find a₁₇ and sum of first 17 terms. Answer: a₁₇ = 131, S₁₇ = 1139 Solution: Identify the first term a₁ = 3. Find the common difference d = 11 - 3 = 8. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 3. Step 2: Find the common difference d = 11 - 3 = 8. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 17: a₁₇ = 3 + (17-1)×8 = 3 + 16×8 = 3 + 128 = 131. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 17: S₁₇ = 17(3 + 131)/2 = 17×134/2 = 2278/2 = 1139. Therefore, a₁₇ = 131 and S₁₇ = 1139.