Question

Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, −1 , . . .(1 point)
Responses

a1=7, an=−2an−1
a subscript 1 baseline equals 7 , a subscript n baseline equals negative 2 a subscript n minus 1 baseline

a1=7, an=an−1−2
a subscript 1 baseline equals 7 , a subscript n baseline equals a subscript n minus 1 baseline minus 2

a1=7, an=an−1+2
a subscript 1 baseline equals 7 , a subscript n baseline equals a subscript n minus 1 baseline plus 2

a1=7, an=2an−1
a subscript 1 baseline equals 7 , a subscript n baseline equals 2 a subscript n minus 1 baseline

Answer 1

To determine the recursive formula for the given arithmetic sequence \( 7, 5, 3, 1, -1, \ldots \), we first observe the pattern:

- The first term \( a_1 = 7 \).
- The second term \( a_2 = 5 \).
- The third term \( a_3 = 3 \).
- The fourth term \( a_4 = 1 \).
- The fifth term \( a_5 = -1 \).

To find the common difference \( d \):
- \( 5 - 7 = -2 \)
- \( 3 - 5 = -2 \)
- \( 1 - 3 = -2 \)
- \( -1 - 1 = -2 \)

The common difference for this sequence is \( -2 \).

Now, the recursive form of an arithmetic sequence is given by:
- The first term: \( a_1 = 7 \)
- The recursive relation: \( a_n = a_{n-1} + d \)

Since \( d = -2 \), we can write the recursive formula as:
\[ a_n = a_{n-1} - 2 \]

Thus, the correct answer is:
**\( a_1 = 7, a_n = a_{n-1} - 2 \)**