Arithmetic Sequences

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Answer Key & Explanations

Arithmetic Sequences · Grade 12 · Worksheet 3

1. Matiu is creating a visual art piece using square tiles arranged in concentric square rings. The innermost ring (Ring 1) has 4 tiles (one on each side of a 2×2 square). Ring 2 has 12 tiles forming the next square border around Ring 1. Ring 3 has 20 tiles, and so on, forming an arithmetic sequence of the number of tiles in each ring. If Matiu continues this pattern until he has 10 complete rings, what is the total number of tiles used in the entire artwork? Answer: 400 Solution: Identify the arithmetic sequence. Ring 1: 4 tiles, Ring 2: 12 tiles, Ring 3: 20 tiles. So a_1 = 4, a_2 = 12, a_3 = 20.
   Full step-by-step solution

   Step 1: Identify the arithmetic sequence. Ring 1: 4 tiles, Ring 2: 12 tiles, Ring 3: 20 tiles. So a_1 = 4, a_2 = 12, a_3 = 20. Step 2: Find the common difference d = a_2 - a_1 = 12 - 4 = 8. Check: a_3 - a_2 = 20 - 12 = 8. So d = 8. Step 3: Find the number of tiles in Ring 10 using a_n = a_1 + (n-1)d. Here n = 10, a_1 = 4, d = 8. So a_10 = 4 + (10-1)*8 = 4 + 9*8 = 4 + 72 = 76. Step 4: Find the total number of tiles in all 10 rings using S_n = n(a_1 + a_n)/2. Here n = 10, a_1 = 4, a_10 = 76. So S_10 = 10(4 + 76)/2 = 10*80/2 = 800/2 = 400. The answer is 400.

2. 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.

3. 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.

4. Sequence: 3, 11, 19, 27... Find a₁₃ and sum of first 13 terms. Answer: a₁₃ = 99, S₁₃ = 663 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 = 13: a₁₃ = 3 + (13-1)×8 = 3 + 12×8 = 3 + 96 = 99. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 13: S₁₃ = 13(3 + 99)/2 = 13×102/2 = 1326/2 = 663. Therefore, a₁₃ = 99 and S₁₃ = 663.

5. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = 80te^(-0.2t) mg/L, where t is measured in hours. The company needs to determine the maximum concentration of the drug and the time at which it occurs to establish proper dosing guidelines. Find both the time when maximum concentration occurs and the maximum concentration value. Answer: t = 5 hours, C = 400/e mg/L (approximately 147.15 mg/L) Solution: To find the maximum concentration, we need to find the critical points of the function C(t) = 80t e^(-0.2t). This is done by taking the derivative, setting it equal to zero, and solving for t.
   Full step-by-step solution

   To find the maximum concentration, we need to find the critical points of the function C(t) = 80t e^(-0.2t). This is done by taking the derivative, setting it equal to zero, and solving for t. Step 1: Find the derivative of C(t). We use the product rule. Let f(t) = 80t and g(t) = e^(-0.2t). The derivative of f(t) is f'(t) = 80. The derivative of g(t) is g'(t) = -0.2 e^(-0.2t) (using the chain rule). The product rule states: (f g)' = f' g + f g'. So, C'(t) = (80) * (e^(-0.2t)) + (80t) * (-0.2 e^(-0.2t)). Simplify this expression: C'(t) = 80 e^(-0.2t) - 16t e^(-0.2t). Factor out the common term, which is 16 e^(-0.2t): C'(t) = 16 e^(-0.2t) (5 - t). Step 2: Set the derivative equal to zero and solve for t. We set C'(t) = 0. 16 e^(-0.2t) (5 - t) = 0. Since 16 is a constant and e^(-0.2t) is never zero for any real number t, the only solution comes from setting (5 - t) = 0. Therefore, 5 - t = 0, which gives t = 5 hours. This is the critical point, and from the context of the problem (concentration rises then falls), it is the time of maximum concentration. Step 3: Find the maximum concentration value. Substitute t = 5 back into the original concentration function C(t). C(5) = 80 * (5) * e^(-0.2 * 5). First, calculate the exponent: -0.2 * 5 = -1. So, C(5) = 400 * e^(-1). This can be written as C(5) = 400 / e mg/L. Step 4: Provide the approximate numerical value. The constant e is approximately 2.71828. So, C(5) = 400 / 2.71828 ≈ 147.15 mg/L. Final Answer: The maximum concentration occurs at t = 5 hours. The maximum concentration is 400/e mg/L, which is approximately 147.15 mg/L.

6. Mason is arranging circular tiles in a triangular formation. The first row has 9 tiles, the second row has 14 tiles, the third row has 19 tiles, and so on, forming an arithmetic sequence. If the formation has 12 rows, find the number of tiles in the 12th row and the total number of tiles used in the entire formation. Answer: a12 = 64, S12 = 438 Solution: Identify the first term a1 = 9 and the common difference d = 14 - 9 = 5. Use the nth term formula: an = a1 + (n-1)d. For the 12th row: a12 = 9 + (12-1)*5 = 9 + 11*5 = 9 + 55 = 64.
   Full step-by-step solution

   Step 1: Identify the first term a1 = 9 and the common difference d = 14 - 9 = 5. Step 2: Use the nth term formula: an = a1 + (n-1)d. For the 12th row: a12 = 9 + (12-1)*5 = 9 + 11*5 = 9 + 55 = 64. Step 3: Find the sum of the first 12 terms using Sn = n(a1 + an)/2. S12 = 12*(9 + 64)/2 = 12*73/2 = 876/2 = 438. The answer is a12 = 64 tiles, S12 = 438 tiles.