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Arithmetic Sequences

Grade 12 · Algebra · Worksheet 1

  1. Noah is arranging square tiles in a visual pyramid pattern. The bottom row has 21 tiles, the next row up has 16 tiles, and each subsequent row has 5 fewer tiles than the row below it, forming an arithmetic sequence. How many tiles are in the 6th row from the bottom? What is the total number of tiles in the entire pyramid if it continues until the top row has 1 tile? Answer: ______________
  2. Sequence: 14, 23, 32, 41... Find a₁₈ and sum of first 18 terms. Answer: ______________
  3. Olivia is a financial analyst for a growing startup. The company's monthly profit (in thousands of dollars) follows an arithmetic sequence. In the first month (n=1), the profit was $5,000. The common difference in monthly profit is an increase of $3,000. What is the total profit, in thousands of dollars, that the company earned from the first month through the 20th month? Answer: ______________
  4. Charlotte is a financial planner who is helping a client save for a major purchase. The client plans to deposit money into a savings account each month, with the deposits forming an arithmetic sequence. The first deposit is $12, and each subsequent deposit increases by $7 from the previous month. Charlotte needs to determine the amount of the 17th deposit and the total amount deposited over the first 17 months to present a savings projection to the client. What is the amount of the 17th deposit, and what is the total amount deposited over the first 17 months? Answer: ______________
  5. Charlotte is analyzing a rectangular garden with an arithmetic sequence of stepping stones along its length. The first stone is placed 7 cm from the edge, the second stone is 12 cm from the edge, the third is 17 cm, and so on. There are 22 stones in total. Charlotte needs to know the distance from the edge to the 22nd stone, as well as the total distance covered by all 22 stones combined (i.e., the sum of their distances from the edge). Find the distance to the 22nd stone and the sum of the distances of all 22 stones. Answer: ______________
  6. ∫(3x² - 6x + 2) dx from 0 to 2 = ? Answer: ______________
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Answer Key & Explanations

Arithmetic Sequences · Grade 12 · Worksheet 1

  1. Noah is arranging square tiles in a visual pyramid pattern. The bottom row has 21 tiles, the next row up has 16 tiles, and each subsequent row has 5 fewer tiles than the row below it, forming an arithmetic sequence. How many tiles are in the 6th row from the bottom? What is the total number of tiles in the entire pyramid if it continues until the top row has 1 tile? Answer: a_6 = -4? (Wait, check sequence: row 1 = 21, row 2 = 16, d = -5. For 6th row: a_6 = 21 + (6-1)(-5) = 21 - 25 = -4, which is impossible for tiles. So the pyramid stops before that. The correct interpretation: The sequence from bottom to top is 21, 16, 11, 6, 1. There are 5 rows. Total sum S_5 = (5/2)(21 + 1) = 55. So answer: 6th row does not exist; total tiles = 55.) Solution: Identify the arithmetic sequence. The bottom row (first term) a_1 = 21. Common difference d = -5 (since each row has 5 fewer tiles).
    Full step-by-step solution

    Step 1: Identify the arithmetic sequence. The bottom row (first term) a_1 = 21. Common difference d = -5 (since each row has 5 fewer tiles). Step 2: Find the number of rows. The top row has 1 tile. Use the formula a_n = a_1 + (n-1)d. Set a_n = 1: 1 = 21 + (n-1)(-5). Simplify: 1 = 21 - 5(n-1) => 5(n-1) = 20 => n-1 = 4 => n = 5. So there are 5 rows. Step 3: The 6th row does not exist because the pyramid only has 5 rows. However, to answer the intended question: the number of tiles in each row from bottom to top: row 1 = 21, row 2 = 16, row 3 = 11, row 4 = 6, row 5 = 1. Step 4: Find the total number of tiles. Use the sum formula S_n = n/2 (a_1 + a_n). Here n = 5, a_1 = 21, a_5 = 1. So S_5 = 5/2 (21 + 1) = (5/2)(22) = 5 * 11 = 55. The total number of tiles in the pyramid is 55. The 6th row does not exist as a valid row.

  2. Sequence: 14, 23, 32, 41... Find a₁₈ and sum of first 18 terms. Answer: a₁₈ = 167, S₁₈ = 1629 Solution: Identify the first term a₁ = 14. Find the common difference d = 23 - 14 = 9. Use the nth term formula aₙ = a₁ + (n-1)d.
    Full step-by-step solution

    Step 1: Identify the first term a₁ = 14. Step 2: Find the common difference d = 23 - 14 = 9. Step 3: Use the nth term formula aₙ = a₁ + (n-1)d. For n = 18: a₁₈ = 14 + (18-1)×9 = 14 + 17×9 = 14 + 153 = 167. Step 4: Use the sum formula Sₙ = n(a₁ + aₙ)/2. For n = 18: S₁₈ = 18(14 + 167)/2 = 18×181/2 = 3258/2 = 1629. Therefore, a₁₈ = 167 and S₁₈ = 1629.

  3. Olivia is a financial analyst for a growing startup. The company's monthly profit (in thousands of dollars) follows an arithmetic sequence. In the first month (n=1), the profit was $5,000. The common difference in monthly profit is an increase of $3,000. What is the total profit, in thousands of dollars, that the company earned from the first month through the 20th month? Answer: 670 Solution: Identify the given values. a₁ = 5 (thousand dollars), d = 3 (thousand dollars), n = 20. Find the profit in the 20th month (a₂₀) using the formula aₙ = a₁ + (n-1)d.
    Full step-by-step solution

    Step 1: Identify the given values. a₁ = 5 (thousand dollars), d = 3 (thousand dollars), n = 20. Step 2: Find the profit in the 20th month (a₂₀) using the formula aₙ = a₁ + (n-1)d. a₂₀ = 5 + (20 - 1) * 3 a₂₀ = 5 + (19) * 3 a₂₀ = 5 + 57 a₂₀ = 62 (thousand dollars) Step 3: Calculate the total profit (sum of the first 20 terms) using the formula Sₙ = n(a₁ + aₙ) / 2. S₂₀ = 20 * (5 + 62) / 2 S₂₀ = 20 * 67 / 2 S₂₀ = 1340 / 2 S₂₀ = 670 (thousand dollars) The total profit earned over the first 20 months is $670,000. The answer is 670.

  4. Charlotte is a financial planner who is helping a client save for a major purchase. The client plans to deposit money into a savings account each month, with the deposits forming an arithmetic sequence. The first deposit is $12, and each subsequent deposit increases by $7 from the previous month. Charlotte needs to determine the amount of the 17th deposit and the total amount deposited over the first 17 months to present a savings projection to the client. What is the amount of the 17th deposit, and what is the total amount deposited over the first 17 months? Answer: 124 and 1156 Solution: Identify the given values. First deposit a1 = 12, common difference d = 7, number of terms n = 17. Find the 17th deposit using the formula a_n = a1 + (n - 1)d.
    Full step-by-step solution

    Step 1: Identify the given values. First deposit a1 = 12, common difference d = 7, number of terms n = 17. Step 2: Find the 17th deposit using the formula a_n = a1 + (n - 1)d. a_17 = 12 + (17 - 1) * 7 a_17 = 12 + 16 * 7 a_17 = 12 + 112 a_17 = 124 Step 3: Find the sum of the first 17 deposits using the formula S_n = n(a1 + a_n)/2. S_17 = 17(12 + 124)/2 S_17 = 17(136)/2 S_17 = 2312/2 S_17 = 1156 Step 4: The 17th deposit is $124, and the total deposited over 17 months is $1156. The answer is 124 and 1156.

  5. Charlotte is analyzing a rectangular garden with an arithmetic sequence of stepping stones along its length. The first stone is placed 7 cm from the edge, the second stone is 12 cm from the edge, the third is 17 cm, and so on. There are 22 stones in total. Charlotte needs to know the distance from the edge to the 22nd stone, as well as the total distance covered by all 22 stones combined (i.e., the sum of their distances from the edge). Find the distance to the 22nd stone and the sum of the distances of all 22 stones. Answer: a22 = 112 cm, S22 = 1309 cm Solution: Identify the arithmetic sequence: first term a1 = 7, common difference d = 12 - 7 = 5, number of terms n = 22. Use the nth term formula: an = a1 + (n-1)d. a22 = 7 + (22-1)*5 = 7 + 21*5 = 7 + 105 = 112 cm.
    Full step-by-step solution

    Step 1: Identify the arithmetic sequence: first term a1 = 7, common difference d = 12 - 7 = 5, number of terms n = 22. Step 2: Use the nth term formula: an = a1 + (n-1)d. a22 = 7 + (22-1)*5 = 7 + 21*5 = 7 + 105 = 112 cm. Step 3: Use the sum formula: Sn = n(a1 + an)/2. S22 = 22*(7 + 112)/2 = 22*119/2 = 22*59.5 = 1309 cm. Step 4: The distance to the 22nd stone is 112 cm, and the total sum of distances is 1309 cm.

  6. ∫(3x² - 6x + 2) dx from 0 to 2 = ? Answer: 0 Solution: Find the antiderivative of 3x² - 6x + 2 Antiderivative = (3x³/3) - (6x²/2) + 2x = x³ - 3x² + 2x Evaluate the antiderivative at the upper limit (x = 2) F(2) = (2)³ - 3(2)² + 2(2) = 8 - 12 + 4 = 0 Evaluate the antiderivative at the lower limit (x = 0) F(0) = (0)³ - 3(0)² + 2(0) = 0 - 0 + 0 = 0…
    Full step-by-step solution

    Step 1: Find the antiderivative of 3x² - 6x + 2 Antiderivative = (3x³/3) - (6x²/2) + 2x = x³ - 3x² + 2x Step 2: Evaluate the antiderivative at the upper limit (x = 2) F(2) = (2)³ - 3(2)² + 2(2) = 8 - 12 + 4 = 0 Step 3: Evaluate the antiderivative at the lower limit (x = 0) F(0) = (0)³ - 3(0)² + 2(0) = 0 - 0 + 0 = 0 Step 4: Apply the Fundamental Theorem of Calculus ∫(3x² - 6x + 2) dx from 0 to 2 = F(2) - F(0) = 0 - 0 = 0 The answer is 0.