Question
Use ther recursive formula for the arthimetic sequence: 7, 5, 3, 1, -1...
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GPT-4o mini
The given arithmetic sequence is:
7, 5, 3, 1, -1...
To derive the recursive formula for this sequence, we can observe the following:
1. The first term \( a_1 \) is 7.
2. To find subsequent terms, we notice that each term decreases by 2 from the previous term.
The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 5 - 7 = -2 \]
In general, for an arithmetic sequence, the recursive formula can be expressed as:
\[
a_n = a_{n-1} + d
\]
Using this information, we can define the recursive formula for the sequence as follows:
- **Base case**: \( a_1 = 7 \)
- **Recursive case**: \( a_n = a_{n-1} - 2 \) for \( n > 1 \)
So, the recursive formula for this arithmetic sequence is:
\[
\begin{align*}
a_1 & = 7 \\
a_n & = a_{n-1} - 2 \text{ for } n > 1
\end{align*}
\]
7, 5, 3, 1, -1...
To derive the recursive formula for this sequence, we can observe the following:
1. The first term \( a_1 \) is 7.
2. To find subsequent terms, we notice that each term decreases by 2 from the previous term.
The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 5 - 7 = -2 \]
In general, for an arithmetic sequence, the recursive formula can be expressed as:
\[
a_n = a_{n-1} + d
\]
Using this information, we can define the recursive formula for the sequence as follows:
- **Base case**: \( a_1 = 7 \)
- **Recursive case**: \( a_n = a_{n-1} - 2 \) for \( n > 1 \)
So, the recursive formula for this arithmetic sequence is:
\[
\begin{align*}
a_1 & = 7 \\
a_n & = a_{n-1} - 2 \text{ for } n > 1
\end{align*}
\]
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