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

Question

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

(1 point)
Option #
is the correct recursive formula for the sequence.

Answer 1

To determine the correct recursive equation for the arithmetic sequence 6, 2, -2, -6, ..., we first identify the pattern of the sequence.

The first term \( a_1 = 6 \).

To find the common difference, we can subtract the second term from the first term:

\[ a_2 - a_1 = 2 - 6 = -4 \]

Similarly:

\[ a_3 - a_2 = -2 - 2 = -4 \]

\[ a_4 - a_3 = -6 - (-2) = -4 \]

The common difference is \(-4\).

Now, using the common difference, we can write the recursive formula. The first term is:

\[ a_1 = 6 \]

And each subsequent term is found by subtracting 4 from the previous term:

\[ a_n = a_{n-1} - 4 \]

Thus, the correct recursive equation is:

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