Question

) Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, \(-1\), . . . (1 point)
\(a_1=7\), \(a_n=-2a_{n-1}\)
\(a_1=7\), \(a_n=a_{n-1}+2\)
\(a_1=7\), \(a_n=a_{n-1}-2\)
\(a_1=7\), \(a_n=2a_{n-1}\)

Answer 1

To find the recursive formula for the arithmetic sequence 7, 5, 3, 1, -1, ..., we first identify the first term and the common difference of the sequence.

The first term, \(a_1\), is 7.

Next, we calculate the common difference:
- \(5 - 7 = -2\)
- \(3 - 5 = -2\)
- \(1 - 3 = -2\)
- \(-1 - 1 = -2\)

The common difference is \(-2\).

Now, we can express the recursive formula:
- The first term is given by \(a_1 = 7\).
- Each subsequent term can be expressed as \(a_n = a_{n-1} - 2\).

So the recursive formula for this arithmetic sequence is:
\[
\begin{align*}
a_1 &= 7, \\
a_n &= a_{n-1} - 2 \quad \text{for } n > 1.
\end{align*}
\]

From the given choices, the correct answer is:
\[
a_1=7, \quad a_n=a_{n-1}-2.
\]