Let's find the explicit formula for the given arithmetic sequence: 9.2, 7.4, 5.6, 3.8, 2. We'll derive the first term and the common difference and then fit it into the standard form of an arithmetic sequence.
The sequence appears as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 \\
\hline
a_n & 9.2 & 7.4 & 5.6 & 3.8 & 2 \\
\hline
\end{array}
\][/tex]
First, we identify the elements needed to form the explicit formula for the sequence:
1. The first term [tex]\(a_1\)[/tex]:
[tex]\[
a_1 = 9.2
\][/tex]
2. The common difference [tex]\(d\)[/tex], which is the difference between consecutive terms:
[tex]\[
d = a_2 - a_1 = 7.4 - 9.2 = -1.8
\][/tex]
The explicit formula for an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
Plugging in the values we've identified:
[tex]\[
a_n = 9.2 + (n-1)(-1.8)
\][/tex]
Let's elaborate on the formula provided in each option and verify them:
1. Option 1: [tex]\(a_n = 1 + 1.8(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 1 + 1.8(1-1) = 1 + 0 = 1
\][/tex]
- The first term should be 9.2, so this option is incorrect.
2. Option 2: [tex]\(a_n = 2 + 1.8(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 2 + 1.8(1-1) = 2 + 0 = 2
\][/tex]
- Again, the first term should be 9.2, so this option is incorrect.
3. Option 3: [tex]\(a_n = 9.2 + (-1.8)(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2
\][/tex]
- Verifying for [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11.0
\][/tex]
- This does not match the second term of the sequence, so this option is incorrect.
4. Option 4: [tex]\(a_n = 9.2 + (-1.8)(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2
\][/tex]
- Substituting [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4
\][/tex]
- Substituting [tex]\(n = 3\)[/tex]:
[tex]\[
a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6
\][/tex]
- Substituting [tex]\(n = 4\)[/tex]:
[tex]\[
a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8
\][/tex]
- Substituting [tex]\(n = 5\)[/tex]:
[tex]\[
a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2.0
\][/tex]
Thus, Option 4 correctly describes the explicit formula for the sequence.
Let's summarize:
The correct explicit formula for the given arithmetic sequence is:
[tex]\[
a_n = 9.2 + (-1.8)(n-1)
\][/tex]
So, the correct option is:
[tex]\[
a_n = 9.2 + (-1.8)(n-1)
\][/tex]
