ACT Math Sequences Series and Matrices Quiz

In an arithmetic sequence, each term is found by adding a constant difference to the previous term. Here, the common difference is 4 (7 - 3). To find the 7th term, use the formula: 3 + (7-1) * 4 = 3 + 24 = 27. Thus, the 7th term is 27.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio. In this sequence, dividing the second term (6) by the first term (2) gives 3. Similarly, dividing the third term (18) by the second term (6) also yields 3, confirming that the common ratio is 3.

To find the sum of the first 10 terms of an arithmetic sequence, use the formula \( S_n = \frac{n}{2} \times (2a + (n-1)d) \). Here, \( n = 10 \), \( a = 5 \), and \( d = 3 \). Plugging in these values gives \( S_{10} = \frac{10}{2} \times (2 \times 5 + 9 \times 3) = 5 \times 32 = 160 \). Thus, the sum is 195.

The sum of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 8 \) and \( r = \frac{1}{2} \). Thus, \( S = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16 \).

The sequence doubles each term: 5, 10 (5 x 2), 20 (10 x 2), 40 (20 x 2). Continuing this pattern, the next term is 80 (40 x 2), followed by 160 (80 x 2). Thus, the 6th term in the sequence is 160.

To find the determinant of the 2x2 matrix A = [[1, 2], [3, 4]], use the formula det(A) = ad - bc, where a, b, c, and d are the elements of the matrix. Here, a = 1, b = 2, c = 3, and d = 4. Thus, det(A) = (1)(4) - (2)(3) = 4 - 6 = -2.

To find the 5th term of the sequence defined by \( a_n = 2n² - 1 \), substitute \( n = 5 \) into the formula. This gives \( a_5 = 2(5²) - 1 = 2(25) - 1 = 50 - 1 = 49 \). Thus, the 5th term is 49.

The series is an arithmetic sequence where the first term is 1 and the common difference is 3. To find the sum, first determine the number of terms. The last term is 31, which can be expressed as 1 + (n-1) * 3. Solving for n gives 11 terms. The sum of an arithmetic series is given by n/2 * (first term + last term), resulting in 165.

In matrix multiplication, the resulting matrix's dimensions are determined by the outer dimensions of the matrices being multiplied. Here, matrix A is 2×3 and matrix B is 3×2. The number of rows from A (2) and the number of columns from B (2) gives the resulting matrix size of 2×2.

In an arithmetic sequence, the common difference can be found using the formula for the nth term: \( a_n = a_1 + (n-1)d \). Here, the first term \( a_1 = 12 \) and the fifth term \( a_5 = 28 \). By substituting these values into the formula, we can solve for \( d \) and find that the common difference is 4.

The sequence provided is the Fibonacci sequence, where each term is the sum of the two preceding terms. The first four terms are 1, 1, 2, and 3. Thus, the 4th term of the sequence is 3, as it follows from the sum of the two previous terms (1 + 2 = 3).

To find the first term of the arithmetic series, we can evaluate the sum formula \( S_n \) at \( n = 1 \). Substituting \( n = 1 \) into \( S_n = 3n² + 2n \) gives \( S_1 = 5 \). Since the sum of the first term is equal to the first term itself, the first term is 5.