Arithmetic Sequences

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Arithmetic Sequences · Grade 12 · Worksheet 3

1. Arithmetic sequence: a₁=7, d=9. Write explicit formula aₙ = ? Answer: aₙ = 9n - 2 Solution: Start with the general formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 7 and d = 9 aₙ = 7 + (n-1)×9 Distribute the 9: aₙ = 7 + 9n - 9 Combine like terms: aₙ = 9n - 2 The explicit formula is aₙ = 9n - 2.
   Full step-by-step solution

   Step 1: Start with the general formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Step 2: Substitute the given values: a₁ = 7 and d = 9 Step 3: aₙ = 7 + (n-1)×9 Step 4: Distribute the 9: aₙ = 7 + 9n - 9 Step 5: Combine like terms: aₙ = 9n - 2 The explicit formula is aₙ = 9n - 2.

2. Arithmetic sequence: a₁=8, d=5. Write explicit formula aₙ = ? Answer: aₙ = 5n + 3 Solution: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 8 and d = 5 aₙ = 8 + (n-1)5 Distribute the 5: aₙ = 8 + 5n - 5 Combine like terms: aₙ = 5n + 3 The explicit formula is aₙ = 5n + 3
   Full step-by-step solution

   Step 1: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d Step 2: Substitute the given values: a₁ = 8 and d = 5 Step 3: aₙ = 8 + (n-1)5 Step 4: Distribute the 5: aₙ = 8 + 5n - 5 Step 5: Combine like terms: aₙ = 5n + 3 The explicit formula is aₙ = 5n + 3

3. A geometric pattern is formed by arranging equilateral triangles in a sequence. The first triangle has side length 2 cm. Each subsequent triangle has side lengths that are 1.5 times the side length of the previous triangle. If this pattern continues infinitely, what is the total area covered by all the triangles? (Area of an equilateral triangle = (√3/4) × side²) Answer: 4√3 Solution: When dealing with an infinite series of geometric shapes where each subsequent shape is a scaled version of the previous one, the areas form a geometric series. The sum of an infinite geometric series exists when the common ratio between consecutive terms has an absolute value less than 1.
   Full step-by-step solution

   When dealing with an infinite series of geometric shapes where each subsequent shape is a scaled version of the previous one, the areas form a geometric series. The sum of an infinite geometric series exists when the common ratio between consecutive terms has an absolute value less than 1. The formula for the sum is the first term divided by (1 minus the common ratio).

4. Given the arithmetic sequence: 15, 25, 35, 45, ... Find the 12th term using the explicit formula aₙ = a₁ + (n-1)d. Answer: 125 Solution: Identify the first term a₁ = 15 Find the common difference d = 25 - 15 = 10 Use the explicit formula aₙ = a₁ + (n-1)d Substitute n = 12, a₁ = 15, d = 10 a₁₂ = 15 + (12-1)×10 = 15 + 11×10 = 15 + 110 = 125 The 12th term is 125.
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 15 Step 2: Find the common difference d = 25 - 15 = 10 Step 3: Use the explicit formula aₙ = a₁ + (n-1)d Step 4: Substitute n = 12, a₁ = 15, d = 10 Step 5: a₁₂ = 15 + (12-1)×10 = 15 + 11×10 = 15 + 110 = 125 The 12th term is 125.

5. During a training program, Maya runs 600 meters on the first day and increases the distance by a constant amount each day. On day 5, Maya runs 1904 meters. What is the daily increase in distance? Answer: 326 Solution: This is an arithmetic sequence with first term a₁ = 600 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 = 600 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: 1904 = 600 + (5 - 1)d Step 4: Subtract 600 from both sides: 1904 - 600 = (5 - 1)d Step 5: Divide by (5 - 1): d = (1904 - 600) / (5 - 1) Step 6: Calculate: d = (1904 - 600) // (5 - 1) = 326 The daily increase is 326 meters.

6. A geometric pattern is formed by arranging circles in a triangular array. The top row has 1 circle, the second row has 3 circles, the third row has 5 circles, and this pattern continues. If the array has 15 rows, what is the total number of circles in the arrangement? Answer: 225 Solution: Identify the pattern. Row 1 has 1 circle, row 2 has 3 circles, row 3 has 5 circles. This forms an arithmetic sequence: 1, 3, 5, 7, ...
   Full step-by-step solution

   Step 1: Identify the pattern. Row 1 has 1 circle, row 2 has 3 circles, row 3 has 5 circles. This forms an arithmetic sequence: 1, 3, 5, 7, ... Step 2: The number of circles in row n is given by: a_n = 1 + (n-1)*2 = 2n - 1. Step 3: We need the total number of circles in 15 rows, which is the sum of the first 15 odd numbers. Step 4: The sum of the first n odd numbers is n^2. Step 5: For n = 15 rows, total circles = 15^2 = 225. The answer is 225.

7. Mere's arithmetic sequence: a₁=12, d=6. Write the explicit formula aₙ = ? Answer: aₙ = 6n + 6 Solution: Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d Substitute the given values: a₁ = 12, d = 6 aₙ = 12 + (n-1)×6 Distribute the 6: aₙ = 12 + 6n - 6 Combine like terms: aₙ = 6n + 6 The explicit formula is aₙ = 6n + 6.
   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 = 12, d = 6 Step 3: aₙ = 12 + (n-1)×6 Step 4: Distribute the 6: aₙ = 12 + 6n - 6 Step 5: Combine like terms: aₙ = 6n + 6 The explicit formula is aₙ = 6n + 6.

8. A geometric pattern is formed by stacking squares in a pyramid arrangement. The bottom row has 10 squares, the row above has 9 squares, and this pattern continues until the top row has 1 square. What is the total number of squares in this pyramid? Answer: 55 Solution: We are told the bottom row has 10 squares, the row above has 9 squares, and so on, until the top row has 1 square. We need the total number of squares, which is the sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
   Full step-by-step solution

   We are told the bottom row has 10 squares, the row above has 9 squares, and so on, until the top row has 1 square. So the number of squares in each row, from top to bottom, is: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. We need the total number of squares, which is the sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. Step 1: Recognize this is an arithmetic series with first term a = 1, last term L = 10, and number of terms n = 10. Step 2: Use the formula for the sum of an arithmetic series: Sum = n * (a + L) / 2. Step 3: Substitute the values: n = 10, a = 1, L = 10. Sum = 10 * (1 + 10) / 2. Step 4: Calculate inside parentheses: 1 + 10 = 11. Step 5: Multiply: 10 * 11 = 110. Step 6: Divide by 2: 110 / 2 = 55. So the total number of squares is 55. We can also check by pairing terms: (1 + 10) = 11, (2 + 9) = 11, (3 + 8) = 11, (4 + 7) = 11, (5 + 6) = 11. That’s 5 pairs, each summing to 11, so 5 * 11 = 55. Both methods confirm the total is 55.