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. An arithmetic sequence has a first term a₁ = -4 and a common difference d = 6. What is the sum of the 11th, 12th, and 13th terms of the sequence? a) 56 b) 186 c) 156 d) 196 First, find the 11th term using the formula for the nth term of an arithmetic sequence:a₁₁ = a₁ + (n - 1)dIn this case, a₁ = -4, d = 6, and n = 11:a₁₁ = -4 + (11 - 1)6 a₁₁ = -4 + 10 * 6 a₁₁ = 56Next, find the 12th and 13th terms using the same formula:a₁₂ = a₁ + (12 - 1)d = -4 + 11 * 6 = 62 a₁₃ = a₁ + (13 - 1)d = -4 + 12 * 6 = 68Now, find the sum of the 11th, 12th, and 13th terms:Sum = a₁₁ + a₁₂ + a₁₃ = 56 + 62 + 68 = 186So, the sum of the 11th, 12th, and 13th terms of the sequence is 186. 3 / 5 3. Given a geometric sequence with the first term a₁ = 1 and the 4th term a₄ = -27, what is the common ratio r? a) -2 b) -3 c) 3 d) 2 Since we know the first term (a₁) and the 4th term (a₄), we can use the formula for a geometric sequence to find the common ratio: aₙ = a₁ * r^(n - 1) We are given: a₁ = 1, a₄ = -27, and n = 4. Plugging in the values: -27 = 1 * r^(4 - 1) -27 = r^3 Now we solve for r: r = -3 So, the common ratio r is -3. The correct answer is -3. 4 / 5 4. Given an arithmetic sequence with the first term a₁ = 7 and a common difference of d = -3, what is the sum of the first 9 terms? a) -9 b) 180 c) -45 d) -57 To find the sum of the terms in a finite arithmetic sequence, we can use the formula: Sₙ = n * (a₁ + aₙ) / 2 Since we are given d = -3, the formula for the nth term of an arithmetic sequence is: aₙ = a₁ + (n - 1)d We are given: a₁ = 7, d = -3, and n = 9. Plugging in the values: a₉ = 7 + (9 - 1)(-3) a₉ = 7 - 24 a₉ = -17 Now we can calculate the sum: S₉ = 9 * (7 + (-17)) / 2 S₉ = 9 * (-10) / 2 S₉ = -45 So, the sum of the first 9 terms of this arithmetic sequence is -45. The correct answer is -45. 5 / 5 5. In a geometric sequence, if the first term is a₁ = 3 and the common ratio r = 2, what is the 4th term of the sequence? a) 16 b) 20 c) 24 d) 28 To find the 4th term of the geometric sequence, we can use the formula: aₙ = a₁ * r^(n-1) We are given: a₁ = 3, r = 2, and n = 4. Plugging in the values: a₄ = 3 * 2^(4-1) a₄ = 3 * 2^3 a₄ = 3 * 8 a₄ = 24 So, the 4th term of the geometric sequence is 24. The correct answer is 24. Your score is 0% Restart Quiz