Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 1

1. 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

2. Tane is constructing a visual pattern using hexagonal tiles arranged in rows. The first row has 9 tiles, the second row has 15 tiles, the third row has 21 tiles, and the fourth row has 27 tiles. If this arithmetic pattern continues, and there are 17 rows in total, what is the total number of hexagonal tiles Tane needs to complete the entire pattern? Answer: 969 Solution: Identify the sequence: 9, 15, 21, 27, ... This is an arithmetic sequence because the difference between consecutive terms is constant. Find the common difference: d = 15 - 9 = 6.
   Full step-by-step solution

   Step 1: Identify the sequence: 9, 15, 21, 27, ... This is an arithmetic sequence because the difference between consecutive terms is constant. Step 2: Find the common difference: d = 15 - 9 = 6. Step 3: First term a1 = 9. Number of terms n = 17. Step 4: Find the last term a17 using the explicit formula: a_n = a1 + (n-1)d = 9 + (17-1)*6 = 9 + 16*6 = 9 + 96 = 105. Step 5: Use the sum formula for an arithmetic series: S_n = n/2 * (a1 + a_n) = 17/2 * (9 + 105) = 17/2 * 114 = 17 * 57 = 969. Step 6: The total number of hexagonal tiles is 969.

3. 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 is given by the function C(t) = 50te^(-0.2t), where t is time in hours after administration. The company needs to determine the maximum concentration of the drug and the time at which this maximum occurs. Find both the time of maximum concentration and the maximum concentration value. Answer: t = 5 hours, C = 250/e ≈ 91.97 mg/L Solution: C(t) = 50 * t * e^(-0.2 * t) We want to find the maximum concentration and the time t at which it occurs. Since C(t) is a product of two functions of t, we use the product rule: Let u(t) = 50t and v(t) = e^(-0.2t).
   Full step-by-step solution

   We are given the concentration function: C(t) = 50 * t * e^(-0.2 * t) We want to find the maximum concentration and the time t at which it occurs. --- **Step 1: Find the derivative of C(t)** Since C(t) is a product of two functions of t, we use the product rule: Let u(t) = 50t and v(t) = e^(-0.2t). Derivative of u: u'(t) = 50 Derivative of v: v'(t) = -0.2 * e^(-0.2t) Product rule: C'(t) = u'(t) * v(t) + u(t) * v'(t) C'(t) = 50 * e^(-0.2t) + 50t * (-0.2 * e^(-0.2t)) Simplify: C'(t) = 50 e^(-0.2t) - 10t e^(-0.2t) C'(t) = e^(-0.2t) * (50 - 10t) --- **Step 2: Set derivative equal to zero to find critical points** e^(-0.2t) is never zero, so we solve: 50 - 10t = 0 10t = 50 t = 5 hours So the critical point is at t = 5 hours. --- **Step 3: Verify it is a maximum** We can check the sign of C'(t) around t = 5: For t < 5, say t = 4: C'(4) = e^(-0.8) * (50 - 40) = positive For t > 5, say t = 6: C'(6) = e^(-1.2) * (50 - 60) = negative Since derivative changes from positive to negative, t = 5 is a maximum. --- **Step 4: Compute maximum concentration** C(5) = 50 * 5 * e^(-0.2 * 5) C(5) = 250 * e^(-1) C(5) = 250 / e Numerically: e ≈ 2.71828 C(5) ≈ 250 / 2.71828 ≈ 91.97 mg/L --- **Final Answer:** Time of maximum concentration: t = 5 hours Maximum concentration: C = 250/e ≈ 91.97 mg/L

4. Arithmetic sequence: a₁=13, d=5. Write explicit formula aₙ = ? Answer: aₙ = 13 + (n-1)×5 Solution: Identify the given values: first term a₁ = 13, common difference d = 5 Use the explicit formula for arithmetic sequences: aₙ = a₁ + (n-1)×d Substitute the given values: aₙ = 13 + (n-1)×5 The explicit formula is aₙ = 13 + (n-1)×5
   Full step-by-step solution

   Step 1: Identify the given values: first term a₁ = 13, common difference d = 5 Step 2: Use the explicit formula for arithmetic sequences: aₙ = a₁ + (n-1)×d Step 3: Substitute the given values: aₙ = 13 + (n-1)×5 Step 4: The explicit formula is aₙ = 13 + (n-1)×5

5. Given the arithmetic sequence: 20, 35, 50, 65, ... Find the 12th term. Answer: 185 Solution: Identify the first term (a₁) and the common difference (d). The first term a₁ = 20. The common difference d = 35 - 20 = 15.
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and the common difference (d). The first term a₁ = 20. The common difference d = 35 - 20 = 15. Step 2: Write the explicit formula for the nth term of an arithmetic sequence: aₙ = a₁ + (n - 1)d. Step 3: Substitute the known values to find the 12th term (a₁₂). a₁₂ = 20 + (12 - 1) * 15 a₁₂ = 20 + (11) * 15 a₁₂ = 20 + 165 a₁₂ = 185 The 12th term is 185.

6. A geometric pattern is formed by arranging squares in a spiral around a central point. The first layer has 8 squares, the second layer has 16 squares, and the third layer has 24 squares. If this pattern continues, how many total squares are there in the first 7 layers? Answer: 224 Solution: Identify the pattern in the number of squares per layer. Layer 1: 8 squares Layer 2: 16 squares Layer 3: 24 squares 8, 16, 24, ... increases by 8 each time.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Identify the pattern in the number of squares per layer.** The problem says: Layer 1: 8 squares Layer 2: 16 squares Layer 3: 24 squares We see that: 8, 16, 24, ... increases by 8 each time. So the number of squares in layer \( n \) is: \( 8 \times n \) Check: Layer 1: \( 8 \times 1 = 8 \) ✓ Layer 2: \( 8 \times 2 = 16 \) ✓ Layer 3: \( 8 \times 3 = 24 \) ✓ --- **Step 2: Write out the number of squares for layers 1 to 7.** Layer 1: \( 8 \times 1 = 8 \) Layer 2: \( 8 \times 2 = 16 \) Layer 3: \( 8 \times 3 = 24 \) Layer 4: \( 8 \times 4 = 32 \) Layer 5: \( 8 \times 5 = 40 \) Layer 6: \( 8 \times 6 = 48 \) Layer 7: \( 8 \times 7 = 56 \) --- **Step 3: Find the total number of squares in 7 layers.** We need: \( 8 + 16 + 24 + 32 + 40 + 48 + 56 \) We can add step by step: \( 8 + 16 = 24 \) \( 24 + 24 = 48 \) \( 48 + 32 = 80 \) \( 80 + 40 = 120 \) \( 120 + 48 = 168 \) \( 168 + 56 = 224 \) --- **Step 4: Verify using the sum of an arithmetic sequence.** The sequence of squares per layer: \( 8, 16, 24, ..., 56 \) This is arithmetic with first term \( a_1 = 8 \), common difference \( d = 8 \), number of terms \( n = 7 \). Sum formula: \( S_n = \frac{n}{2} \times (a_1 + a_n) \) Here: \( a_7 = 8 \times 7 = 56 \) \( S_7 = \frac{7}{2} \times (8 + 56) \) \( S_7 = \frac{7}{2} \times 64 \) \( S_7 = 7 \times 32 = 224 \) --- **Final Answer:** 224

7. Emma'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 So d = 6 a_n = a_1 + (n - 1)d Where a_1 = 7, d = 6, n = 15 a_15 = 7 + (15 - 1) × 6 a_15 = 7 + (14) × 6 a_15 = 7 + 84 a_15 = 91 The 15th term is 91.
   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 So d = 6 Step 2: Use the explicit formula for arithmetic sequences a_n = a_1 + (n - 1)d Where a_1 = 7, d = 6, n = 15 Step 3: Substitute the values a_15 = 7 + (15 - 1) × 6 a_15 = 7 + (14) × 6 a_15 = 7 + 84 a_15 = 91 The 15th term is 91.

8. 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.