Question

determine the next 3 terms in the arithmetic sequence 3, 10, 17, 24, \boxed{\quad}, \boxed{\quad}, \boxed{\quad}.

Explanation:

Step1: Find the common difference

In an arithmetic sequence, the common difference \(d\) is found by subtracting consecutive terms. Let's take the second term minus the first term: \(10 - 3 = 7\), the third term minus the second term: \(17 - 10 = 7\), and the fourth term minus the third term: \(24 - 17 = 7\). So the common difference \(d = 7\).

Step2: Find the fifth term

To find the next term (fifth term) in the arithmetic sequence, we use the formula for the \(n\)-th term of an arithmetic sequence \(a_n=a_{n - 1}+d\), where \(a_{n-1}\) is the previous term and \(d\) is the common difference. Here, \(a_4 = 24\) and \(d=7\), so \(a_5=a_4 + d=24 + 7=31\).

Step3: Find the sixth term

Using the same formula, with \(a_5 = 31\) and \(d = 7\), we get \(a_6=a_5 + d=31+7 = 38\).

Step4: Find the seventh term

Using the formula again, with \(a_6 = 38\) and \(d = 7\), we get \(a_7=a_6 + d=38 + 7=45\).

Answer:

The next three terms are \(31\), \(38\), and \(45\).