Given the recursive formula for the arithmetic sequence a1=−5, an=an−1+4, find the first three terms of the sequence.(2 points)

Answer 1

To find the first three terms of the arithmetic sequence defined by the recursive formula \( a_1 = -5 \) and \( a_n = a_{n-1} + 4 \), we can calculate the terms step by step:

1. The first term is given: \[ a_1 = -5 \]

2. To find the second term \( a_2 \): \[ a_2 = a_1 + 4 = -5 + 4 = -1 \]

3. To find the third term \( a_3 \): \[ a_3 = a_2 + 4 = -1 + 4 = 3 \]

Now we have the first three terms of the sequence:

• \( a_1 = -5 \)
• \( a_2 = -1 \)
• \( a_3 = 3 \)

Thus, the first three terms of the sequence are \(-5\), \(-1\), and \(3\).

Answer 2

how do you come up with the equations?

Answer 3

The equations for the arithmetic sequence are derived from the characteristics of arithmetic sequences and the given recursive formula. Here’s how the process works:

1. Understanding Arithmetic Sequences:

   • An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is often referred to as the "common difference" (\(d\)).
   • For any term in the sequence, you can express it in terms of the previous term by adding the common difference.

2. Given Information:

   • The problem states that \(a_1 = -5\) (the first term) and provides a recursive formula \(a_n = a_{n-1} + 4\). Here, the common difference \(d\) is \(4\). This means that each term in the sequence is obtained by adding \(4\) to the preceding term.

3. Using the Recursive Formula:

   • Start with the first term: \[ a_1 = -5 \]
   • To find the second term, apply the recursive relation: \[ a_2 = a_1 + 4 \]
     • Substitute \(a_1\): \[ a_2 = -5 + 4 = -1 \]
   • For the third term, use the same approach: \[ a_3 = a_2 + 4 \]
     • Substitute \(a_2\): \[ a_3 = -1 + 4 = 3 \]

4. Generalization:

   • From the recursive definition, you can derive a formula for the \(n\)-th term of the arithmetic sequence: \[ a_n = a_1 + (n - 1)d \]
   • In this case, it would be: \[ a_n = -5 + (n - 1) \cdot 4 \]
   • This general term can be verified by plugging in specific values of \(n\).

By following these steps, you can systematically calculate the terms of the sequence based on the initial condition and the recursive relation provided in the problem.