### Step-by-Step Detailed Solution:
#### 1. Write an Arithmetic Sequence in Explicit Form and a Linear Function in Slope-Intercept Form for Each Option
To solve this, let's identify the key elements for each option:
- Option 1: [tex]$18/month
- Explicit Form: \(a_n = 18n\)
- Slope-Intercept Form: \(y = 18x\)
- Option 2: $[/tex]10 to initiate, then [tex]$15/month
- Explicit Form: \(a_n = 10 + 15n\)
- Slope-Intercept Form: \(y = 15x + 10\)
- Option 3: $[/tex]75 to initiate, then [tex]$5/month
- Explicit Form: \(a_n = 75 + 5n\)
- Slope-Intercept Form: \(y = 5x + 75\)
Here is the solution formatted in a table:
\[
\begin{array}{|c|c|c|}
\hline
\text{Option} & \text{Explicit Form} & \text{Slope-Intercept Form} \\
\hline
\text{Option 1} & a_n = 18n & y = 18x \\
\hline
\text{Option 2} & a_n = 10 + 15n & y = 15x + 10 \\
\hline
\text{Option 3} & a_n = 75 + 5n & y = 5x + 75 \\
\hline
\end{array}
\]
#### 2. How is the Explicit Form of Arithmetic Sequences and the Slope-Intercept Form for a Linear Function Similar?
The explicit form of arithmetic sequences and the slope-intercept form of linear functions are similar in that both formats describe a linear relationship between variables.
In an arithmetic sequence:
- The formula \(a_n = a + (n-1)d\) (where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number) simplifies to \(a_n = a_1 + dn\) if we adjust indexing.
In a slope-intercept form for a linear function:
- The formula \(y = mx + b\) identifies \(m\) as the slope (rate of change) and \(b\) as the y-intercept (starting value).
Essentially, both forms involve an initial value/start plus a consistent rate of change over time.
#### 3. Explain What the \(y\)-Intercept and Slope Mean for Each of the Options
Let's interpret the \(y\)-intercept and slope for each of the options:
- Option 1:
- \(y\)-Intercept: 0 (There is no initial cost; the cost starts from zero.)
- Slope: 18 (Each additional month costs $[/tex]18.)
- Option 2:
- [tex]\(y\)[/tex]-Intercept: 10 (The initial cost is [tex]$10.)
- Slope: 15 (Each additional month costs $[/tex]15.)
- Option 3:
- [tex]\(y\)[/tex]-Intercept: 75 (The initial cost is [tex]$75.)
- Slope: 5 (Each additional month costs $[/tex]5.)
By analyzing these values, we can see how different plans fare in terms of initial and recurring costs. The [tex]\(y\)[/tex]-intercept gives you the starting cost without any months counted, and the slope indicates how much the cost increases per month.
