Question

find the recursive formula for the arithmetic sequence 7, 5, 3, 1, -1

Answer 1

To find the recursive formula for the given arithmetic sequence \(7, 5, 3, 1, -1\), we first need to identify the first term and the common difference.

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

2. **Find the common difference (\(d\))**:
The common difference can be calculated by subtracting any term from the subsequent term. Let's calculate it using the first two terms:
\[
d = 5 - 7 = -2
\]
This means that each term is \(2\) less than the previous term.

Now, we can express the recursive formula. The general form of the recursive formula for an arithmetic sequence is:
\[
a_n = a_{n-1} + d
\]
where \(d\) is the common difference.

For this sequence, the recursive formula can be written as:
- \(a_1 = 7\)
- \(a_n = a_{n-1} - 2\) for \(n > 1\)

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