Question

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+(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

An=6(−12)n−1

Answer 1

To identify the explicit formula for the sequence \(12, 6, 0, -6, \ldots\), let's first observe the pattern of 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 \(a_4 = -6\)

From this, we can see that each term decreases by \(6\):
- From \(12\) to \(6\) is \(-6\)
- From \(6\) to \(0\) is \(-6\)
- From \(0\) to \(-6\) is \(-6\)

Thus, the common difference \(d\) is \(-6\). Therefore, this sequence is an arithmetic sequence. The formula for the \(n\)th term of an arithmetic sequence can be expressed as:

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

where:
- \(A_1 = 12\) (the first term)
- \(d = -6\) (the common difference)

Substituting these values into the formula, we get:

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

Thus, the correct explicit formula that represents the sequence is:

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

So, from the options provided, the correct response is:

**An=12+(n−1)(−6)**