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

Question

, −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 determine the correct recursive equation for the arithmetic sequence \(6, 2, -2, -6, \ldots\), we first need to identify two key components: the first term and the common difference.

1. First Term: The first term \(a_1\) is \(6\).

2. Common Difference: To find the common difference, we can subtract consecutive terms:

   • From \(6\) to \(2\): the difference is \(2 - 6 = -4\).
   • From \(2\) to \(-2\): the difference is \(-2 - 2 = -4\).
   • From \(-2\) to \(-6\): the difference is \(-6 - (-2) = -4\).

   Thus, the common difference \(d\) is \(-4\).

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

• The first term is given by \(a_1 = 6\).
• Each subsequent term can be defined using the recursive formula: \(a_n = a_{n-1} + d\), with \(d = -4\).

Therefore, the correct recursive equation becomes:

• \(a_1 = 6\)
• \(a_n = a_{n-1} - 4\)

This matches Option #2:

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