Math Quiz - Sequences Report a question What's wrong with this question? You cannot submit an empty report. Please add some details. Math Quiz - Sequences Covering: arithmetic and geometric sequences, partial sums, and recursive formulas, as well as real-world applications of sequence concepts. 1 / 5 1. A geometric sequence has a first term a₁ = 5 and a 3rd term a₃ = 45. What is the 6th term of the sequence? a) 1215 b) 675 c) 1485 d) 405 Since we know the first term (a₁) and the 3rd term (a₃), we can use the formula for a geometric sequence to find the common ratio: a₃ = a₁ * r^(3-1) We are given: a₁ = 5, a₃ = 45, and n = 3. Plugging in the values: 45 = 5 * r^(2) 9 = r^2 Now we solve for r: r = 3 Now we have the first term and the common ratio, so we can find the 6th term using the formula: aₙ = a₁ * r^(n - 1) We are given: a₁ = 5, r = 3, and n = 6. Plugging in the values: a₆ = 5 * 3^(6 - 1) a₆ = 5 * 3^5 a₆ = 5 * 243 So, the 6th term of the geometric sequence is 1215. The correct answer is A) 1215. 2 / 5 2. For a geometric sequence, given the first term a₁ = 3 and the common ratio r = 2, what is the 5th term of the sequence? a) 36 b) 60 c) 48 d) 24 To compute any term in a geometric sequence, we can use the formula: aₙ = a₁ * r^(n - 1) We are given: a₁ = 3, r = 2, and we need to find the 5th term, so n = 5. Plugging in the values: a₅ = 3 * 2^(5 - 1) a₅ = 3 * 2^4 a₅ = 3 * 16 a₅ = 48 So, the 5th term of the geometric sequence is 48. The correct answer is 48. 3 / 5 3. If the first three terms of an arithmetic sequence are 3, 5, and 7, what is the common difference d? a) 1 b) 2 c) 3 d) 4 To find the common difference (d), we can subtract the first term from the second term, or the second term from the third term: d = 5 - 3 = 2 d = 7 - 5 = 2 So, the common difference d is 2. The correct answer is 2. 4 / 5 4. In a geometric sequence, if the first term is a₁ = 1 and the common ratio r = 4, what is the sum of the first 3 terms of the sequence? a) 17 b) 21 c) 25 d) 29 To find the sum of the first 3 terms of a geometric sequence, we can use the formula: S₃ = a₁ * (1 - r^3) / (1 - r) We are given: a₁ = 1, r = 4. Plugging in the values: S₃ = 1 * (1 - 4^3) / (1 - 4) S₃ = 1 * (1 - 64) / (-3) S₃ = (1 * -63) / (-3) S₃ = 21 So, the sum of the first 3 terms of the geometric sequence is 21. The correct answer is 21. 5 / 5 5. In a geometric sequence, if the first term is a₁ = 5 and the common ratio r = 0.5, what is the 3rd term of the sequence? a) 2 b) 1.25 c) 0.625 d) 3 To find the 3rd term of the geometric sequence, we can use the formula: aₙ = a₁ * r^(n-1) We are given: a₁ = 5, r = 0.5, and n = 3. Plugging in the values: a₃ = 5 * 0.5^(3-1) a₃ = 5 * 0.5^2 a₃ = 5 * 0.25 a₃ = 1.25 So, the 3rd term of the geometric sequence is 1.25. The correct answer is 1.25. Your score is 0% Restart Quiz