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Sagot :
### 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.
#### 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.
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