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Given the sequence 12, 6, 0, −6 , . . . , identify the explicit formula that represents the sequence.(1 point) Responses An=12(−6)n−1 upper A subscript n baseline equals 12 left parenthesis negative 6 right parenthesis superscript n minus 1 baseline An=6(−12)n−1 upper A subscript n baseline equals 6 left parenthesis negative 12 right parenthesis superscript n minus 1 baseline An=−6+(n−1)12 upper A subscript n baseline equals negative 6 plus left parenthesis n minus 1 right parenthesis 12 An=12+(n−1)(−6) upper A subscript n baseline equals 12 plus left parenthesis n minus 1 right parenthesis left parenthesis negative 6 right parenthesis

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To find the explicit formula for the sequence \(12, 6, 0, -6, \ldots\), we first observe the pattern in the sequence.

  1. The first term \(A_1 = 12\).
  2. The second term \(A_2 = 6\).
  3. The third term \(A_3 = 0\).
  4. The fourth term \(A_4 = -6\).

Next, we notice that the sequence decreases by \(6\) with each term:

  • From \(12\) to \(6\), it decreases by \(6\).
  • From \(6\) to \(0\), it decreases by \(6\).
  • From \(0\) to \(-6\), it decreases by \(6\).

The common difference \(d\) is \(-6\). This suggests that the sequence is an arithmetic sequence.

The general formula for an arithmetic sequence is given by:

\[ A_n = A_1 + (n - 1) \cdot d \]

Where:

  • \(A_1\) is the first term (which is \(12\)),
  • \(d\) is the common difference (which is \(-6\)),
  • \(n\) is the term number.

Substituting the values into the formula:

\[ A_n = 12 + (n - 1) \cdot (-6) \]

Simplifying this:

\[ A_n = 12 - 6(n - 1) \] \[ A_n = 12 - 6n + 6 \] \[ A_n = 18 - 6n \]

Thus, the explicit formula for the sequence is:

\[ A_n = 18 - 6n \]

Among the options provided, this corresponds to:

\[ A_n = -6 + (n - 1) \cdot 12 \]

Therefore, the correct response is: \[ \boxed{A_n = -6 + (n - 1) \cdot 12} \]