What is the explicit formula for the arithmetic sequence in the table below?

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$n$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$a_n$ & 9.2 & 7.4 & 5.6 & 3.8 & 2 \\
\hline
\end{tabular}
\][/tex]

A. [tex]\[a_n = 1 + 1.8(n - 1)\][/tex]

B. [tex]\[a_n = 2 + 1.8(1 - n)\][/tex]

C. [tex]\[a_n = 9.2 + (-1.8)(1 - n)\][/tex]

D. [tex]\[a_n = 9.2 + (-1.8)(n - 1)\][/tex]



Answer :

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]