An arithmetic sequence is defined by a common difference between consecutive terms. To find the recursive equation for the given sequence \(6, 2, -2, -6, \ldots\), we first determine the common difference.
Calculating the common difference:
- \(2 - 6 = -4\)
- \(-2 - 2 = -4\)
- \(-6 - (-2) = -4\)
The common difference \(d\) is \(-4\).
The first term of the sequence (denoted as \(a_1\)) is \(6\).
The recursive formula for an arithmetic sequence can be expressed as: \[ a_n = a_{n-1} + d \] where:
- \(a_n\) is the \(n\)-th term,
- \(a_{n-1}\) is the previous term,
- \(d\) is the common difference.
Substituting the values we found:
- Start with the first term: \(a_1 = 6\)
- The common difference: \(d = -4\)
Thus, the recursive equations for this arithmetic sequence are: \[ a_1 = 6 \] \[ a_n = a_{n-1} - 4 \quad \text{for } n \geq 2 \]
So the complete recursive formula is: \[ \begin{align*} a_1 &= 6 \ a_n &= a_{n-1} - 4 \quad \text{for } n \geq 2 \end{align*} \]
1 answer