Question

Which option is the correct recursive equation for the arithmetic sequence 6,2,-2,-6…..

Answer 1

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:
\[
\begin{align*}
a_1 &= 6 \\
a_n &= a_{n-1} - 4 \quad \text{for } n \geq 2
\end{align*}
\]