Question

Which option is the correct recursive equation for the arithmetic sequence 6, 2, −2
, −6
, . . .?

Option #1: a1=6
, an=an−1+4

Option #2: a1=6
, an=an−1−4

Option #3: a1=6
, an=4an−1

Option #4: a1=6
, an=−4an−1

Answer 1

To find the correct recursive equation for the arithmetic sequence 6, 2, -2, -6, ..., we need to determine the first term and the common difference.

1. **Identify the first term:**
- The first term \( a_1 = 6 \).

2. **Calculate the common difference:**
- The difference between the consecutive terms is:
- \( 2 - 6 = -4 \)
- \( -2 - 2 = -4 \)
- \( -6 - (-2) = -4 \)
- The common difference \( d = -4 \).

Using this information, we can define the recursive equation for the sequence:

- The first term is given by \( a_1 = 6 \).
- The recursive formula for the \( n \)-th term is \( a_n = a_{n-1} + d \). Since the common difference \( d = -4 \), we have:
- \( a_n = a_{n-1} - 4 \).

Therefore, the correct option is:

**Option #2: \( a_1 = 6 \), \( a_n = a_{n-1} - 4 \)**.