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

a1=7, an=an−1+2

Answer 1

To find the recursive formula for the given arithmetic sequence 7, 5, 3, 1, -1, ..., we can observe the pattern of the sequence.

1. Identify the first term:
\( a_1 = 7 \)

2. Determine the common difference:
The difference between consecutive terms is:
\( 5 - 7 = -2 \)
\( 3 - 5 = -2 \)
\( 1 - 3 = -2 \)
\( -1 - 1 = -2 \)

So, the common difference \( d = -2 \).

3. Write the recursive formula:
The general form of a recursive formula for an arithmetic sequence is:
\( a_n = a_{n-1} + d \)
In this case, since the common difference \( d = -2 \), we have:
\( a_n = a_{n-1} - 2 \)

Putting this together, the recursive formula is:
- \( a_1 = 7 \)
- \( a_n = a_{n-1} - 2 \)

Thus, the correct response is:

**a1=7, an=an−1−2**
\( a_1 = 7, \quad a_n = a_{n-1} - 2 \)