Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 2

1. Emma is training for a charity run and is increasing her weekly running distance by a constant amount each week. In week 1, she runs 7 km. In week 7, she runs 43 km. Her goal is to run at least 100 km in a single week. Assuming the pattern continues, what is the minimum number of weeks she needs to train to reach or exceed 100 km in a week, and what is her total running distance over all those weeks from week 1 up to and including that week? Answer: Week 17, total 867 km Solution: Identify known values. Week 1: a₁ = 7 km. Week 7: a₇ = 43 km.
   Full step-by-step solution

   Step 1: Identify known values. Week 1: a₁ = 7 km. Week 7: a₇ = 43 km. The sequence is arithmetic, so aₙ = a₁ + (n-1)d. Step 2: Use a₇ to find d: 43 = 7 + (7-1)d → 43 = 7 + 6d → 36 = 6d → d = 6. Step 3: Find n such that aₙ ≥ 100: 100 ≤ 7 + (n-1)6 → 100 ≤ 7 + 6n - 6 → 100 ≤ 6n + 1 → 99 ≤ 6n → n ≥ 16.5. Since n must be a whole number, n = 17. Step 4: Verify a₁₇ = 7 + (17-1)6 = 7 + 16×6 = 7 + 96 = 103 km, which exceeds 100 km. Step 5: Find the total distance from week 1 to week 17 using Sₙ = n(a₁ + aₙ)/2: S₁₇ = 17(7 + 103)/2 = 17(110)/2 = 17×55 = 935 km. The answer is week 17, with a total of 935 km.

2. Mere is constructing a visual pyramid of square tiles. The bottom row has 2 tiles. The second row from the bottom has 4 tiles, the third row from the bottom has 6 tiles, and so on, with each row above having 2 more tiles than the row below it, forming an arithmetic sequence. There are 12 rows in total. What is the total number of tiles in the pyramid? Answer: 156 Solution: Identify the arithmetic sequence. The bottom row has 2 tiles, the next has 4, then 6, and so on. So first term a1 = 2, common difference d = 2, number of terms n = 12.
   Full step-by-step solution

   Step 1: Identify the arithmetic sequence. The bottom row has 2 tiles, the next has 4, then 6, and so on. So first term a1 = 2, common difference d = 2, number of terms n = 12. Step 2: Find the number of tiles in the top row (the 12th term). Use an = a1 + (n - 1)d. a12 = 2 + (12 - 1)*2 = 2 + 11*2 = 2 + 22 = 24. Step 3: Find the total number of tiles using the sum formula Sn = n(a1 + an)/2. S12 = 12*(2 + 24)/2 = 12*26/2 = 12*13 = 156. The answer is 156.

3. A geometric pattern is formed by stacking triangular layers of blocks. The first layer has 1 block, the second layer has 3 blocks arranged in a triangle, the third layer has 6 blocks in a larger triangle, and this pattern continues where each subsequent layer forms a triangular number pattern (1, 3, 6, 10, 15...). If the pattern has n layers, the total number of blocks is given by the sum of the first n triangular numbers. Find the explicit formula for the total number of blocks in terms of n. Answer: n(n+1)(n+2)/6 Solution: Recall that the k-th triangular number is T_k = k(k+1)/2 The total number of blocks after n layers is the sum of the first n triangular numbers: S_n = T_1 + T_2 + ...
   Full step-by-step solution

   Step 1: Recall that the k-th triangular number is T_k = k(k+1)/2 Step 2: The total number of blocks after n layers is the sum of the first n triangular numbers: S_n = T_1 + T_2 + ... + T_n Step 3: Write this as S_n = ∑[k=1 to n] k(k+1)/2 Step 4: Expand the numerator: S_n = (1/2)∑[k=1 to n] (k^2 + k) Step 5: Separate the sum: S_n = (1/2)[∑k^2 + ∑k] from k=1 to n Step 6: Use the formulas: ∑k = n(n+1)/2 and ∑k^2 = n(n+1)(2n+1)/6 Step 7: Substitute: S_n = (1/2)[n(n+1)(2n+1)/6 + n(n+1)/2] Step 8: Factor out n(n+1)/2: S_n = (1/2)[n(n+1)/2][(2n+1)/3 + 1] Step 9: Simplify inside brackets: (2n+1)/3 + 1 = (2n+1+3)/3 = (2n+4)/3 = 2(n+2)/3 Step 10: Now we have: S_n = (1/2)[n(n+1)/2][2(n+2)/3] Step 11: Simplify: S_n = n(n+1)(n+2)/6 The answer is n(n+1)(n+2)/6.

4. Ava is arranging hexagonal tiles in a triangular pattern. The first row has 1 tile, the second row has 6 tiles, the third row has 11 tiles, and the fourth row has 16 tiles. If the pattern continues in this arithmetic sequence, determine the total number of tiles needed to build a structure with 21 rows. Answer: 1071 Solution: Identify the first term and common difference. The sequence is 1, 6, 11, 16, ... So a_1 = 1 and d = 5 (since 6 - 1 = 5, 11 - 6 = 5, etc.).
   Full step-by-step solution

   Step 1: Identify the first term and common difference. The sequence is 1, 6, 11, 16, ... So a_1 = 1 and d = 5 (since 6 - 1 = 5, 11 - 6 = 5, etc.). Step 2: Find the number of tiles in the 21st row using a_n = a_1 + (n-1)d. a_21 = 1 + (21-1) * 5 = 1 + 20 * 5 = 1 + 100 = 101. Step 3: Find the total number of tiles for 21 rows using S_n = n(a_1 + a_n)/2. S_21 = 21 * (1 + 101) / 2 = 21 * 102 / 2 = 21 * 51 = 1071. The answer is 1071.

5. Sequence: 4, 14, 24, 34... Find a₂₀ and sum of first 20 terms. Answer: a₂₀ = 194, S₂₀ = 1980 Solution: Identify the first term a₁ = 4. Find the common difference d = 14 - 4 = 10. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 4. Step 2: Find the common difference d = 14 - 4 = 10. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 20: a₂₀ = 4 + (20-1)×10 = 4 + 19×10 = 4 + 190 = 194. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 20: S₂₀ = 20(4 + 194)/2 = 20×198/2 = 3960/2 = 1980. Therefore, a₂₀ = 194 and S₂₀ = 1980.

6. Sequence: 8, 17, 26, 35... Find a₁₈ and sum of first 18 terms. Answer: a₁₈ = 161, S₁₈ = 1521 Solution: Identify the first term a₁ = 8. Find the common difference d = 17 - 8 = 9. Use the nth term formula aₙ = a₁ + (n-1)d.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 8. Step 2: Find the common difference d = 17 - 8 = 9. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 18: a₁₈ = 8 + (18-1)×9 = 8 + 17×9 = 8 + 153 = 161. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 18: S₁₈ = 18(8 + 161)/2 = 18×169/2 = 3042/2 = 1521. Therefore, a₁₈ = 161 and S₁₈ = 1521.

7. A geometric pattern is formed by stacking squares in a triangular arrangement. The first row contains 1 square, the second row contains 3 squares, the third row contains 5 squares, and this pattern continues. If the pattern has 10 rows, what is the total number of squares used? Answer: 100 Solution: The number of squares in each row follows the sequence: Row 1: 1 Row 2: 3 Row 3: 5 Row 4: 7 ... This is an arithmetic sequence where each term increases by 2.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Understand the pattern** The number of squares in each row follows the sequence: Row 1: 1 Row 2: 3 Row 3: 5 Row 4: 7 ... This is an arithmetic sequence where each term increases by 2. **Step 2: Write the general term** For row n, the number of squares = 1 + (n - 1) * 2 Simplify: 1 + 2n - 2 = 2n - 1 So row n has (2n - 1) squares. **Step 3: List the number of squares for each of the 10 rows** Row 1: 2*1 - 1 = 1 Row 2: 2*2 - 1 = 3 Row 3: 2*3 - 1 = 5 Row 4: 2*4 - 1 = 7 Row 5: 2*5 - 1 = 9 Row 6: 2*6 - 1 = 11 Row 7: 2*7 - 1 = 13 Row 8: 2*8 - 1 = 15 Row 9: 2*9 - 1 = 17 Row 10: 2*10 - 1 = 19 **Step 4: Find the total number of squares** We need to sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 This is the sum of the first 10 odd numbers. **Step 5: Use the formula for the sum of the first n odd numbers** The sum of the first n odd numbers = n^2. So for n = 10, total squares = 10^2 = 100. **Step 6: Verify by pairing if needed** Pair first and last: 1 + 19 = 20 Pair second and second last: 3 + 17 = 20 Similarly: 5 + 15 = 20, 7 + 13 = 20, 9 + 11 = 20 We have 5 pairs, each summing to 20. 5 * 20 = 100. **Final Answer:** 100