To determine the explicit formula for the arithmetic sequence given in the table, let's follow a detailed, step-by-step process.
1. Identify Key Features of the Sequence:
- First few terms of the sequence are: 9.2, 7.4, 5.6, 3.8, 2.
- These terms decrease as [tex]\( n \)[/tex] increases, which suggests a negative common difference.
2. Calculate the Common Difference:
[tex]\[
d = a_2 - a_1 = 7.4 - 9.2 = -1.8
\][/tex]
The common difference [tex]\( d \)[/tex] is [tex]\(-1.8\)[/tex].
3. Identify the First Term:
[tex]\[
a_1 = 9.2
\][/tex]
4. Formulate the General Formula:
The general formula for an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \times d
\][/tex]
Plugging in the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex], we get:
[tex]\[
a_n = 9.2 + (n - 1) \times (-1.8)
\][/tex]
Simplifying, we get:
[tex]\[
a_n = 9.2 - 1.8(n - 1) = 9.2 - 1.8n + 1.8 = 11 - 1.8n
\][/tex]
5. Evaluate the Given Options:
- Option a: [tex]\( a_n = 1 + 1.8(n - 1) \)[/tex]
[tex]\[
\text{For } n = 1, 2, 3, 4, 5:
\][/tex]
[tex]\[
n = 1 \Rightarrow a_1 = 1 + 1.8(1-1) = 1 \quad (\text{not } 9.2)
\][/tex]
- Option b: [tex]\( a_n = 2 + 1.8(1 - n) \)[/tex]
[tex]\[
\text{For } n = 1, 2, 3, 4, 5:
\][/tex]
[tex]\[
n = 1 \Rightarrow a_1 = 2 + 1.8(1-1) = 2 \quad (\text{not } 9.2)
\][/tex]
- Option c: [tex]\( a_n = 9.2 + (-1.8)(1 - n) \)[/tex]
[tex]\[
\text{For } n = 1, 2, 3, 4, 5:
\][/tex]
[tex]\[
n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)}
\][/tex]
[tex]\[
n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11 \quad (\text{not } 7.4)
\][/tex]
- Option d: [tex]\( a_n = 9.2 + (-1.8)(n - 1) \)[/tex]
[tex]\[
\text{For } n = 1, 2, 3, 4, 5:
\][/tex]
[tex]\[
n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)}
\][/tex]
[tex]\[
n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4 \quad \text{(matches)}
\][/tex]
[tex]\[
n = 3 \Rightarrow a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6 \quad \text{(matches)}
\][/tex]
[tex]\[
n = 4 \Rightarrow a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8 \quad \text{(matches)}
\][/tex]
[tex]\[
n = 5 \Rightarrow a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2 \quad \text{(matches)}
\][/tex]
6. Determine the Correct Formula:
After evaluating the given options, we see that the only option that fits the first five terms of the sequence is the formula:
[tex]\[
a_n = 9.2 + (-1.8)(n - 1)
\][/tex]
Therefore, the explicit formula for the given sequence is:
[tex]\[
a_n = 9.2 + (-1.8)(n - 1)
\][/tex]
