Question

fill in the next three terms of the arithmetic sequence: -4, -1, 2, 5, \boxed{\quad}, \boxed{\quad}, \boxed{\quad}.

Explanation:

Step1: Find the common difference

To find the common difference \( d \) in an arithmetic sequence, we subtract consecutive terms. Let's take the second term minus the first term: \( -1 - (-4) = -1 + 4 = 3 \). We can check with the third term minus the second term: \( 2 - (-1) = 2 + 1 = 3 \), and the fourth term minus the third term: \( 5 - 2 = 3 \). So the common difference \( d = 3 \).

Step2: Find the fifth term

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. For the fifth term, \( n = 5 \), \( a_1 = -4 \), \( d = 3 \). So \( a_5 = -4 + (5 - 1) \times 3 = -4 + 4 \times 3 = -4 + 12 = 8 \).

Step3: Find the sixth term

For the sixth term, \( n = 6 \). Using the formula, \( a_6 = -4 + (6 - 1) \times 3 = -4 + 5 \times 3 = -4 + 15 = 11 \).

Step4: Find the seventh term

For the seventh term, \( n = 7 \). Using the formula, \( a_7 = -4 + (7 - 1) \times 3 = -4 + 6 \times 3 = -4 + 18 = 14 \).

Answer:

The next three terms are \( 8 \), \( 11 \), and \( 14 \). So the sequence with the next three terms is: \(-4, -1, 2, 5, \boldsymbol{8}, \boldsymbol{11}, \boldsymbol{14}\)