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Sagot :
Certainly! Let's break down the solution step-by-step for Karla's Gamestation monthly plans using the given values for each option.
### 1. Arithmetic Sequence and Linear Function for Each Option
We need to derive both the arithmetic sequence in explicit form and the linear function in slope-intercept form for each subscription option.
#### Option 1
Details:
- Monthly cost: [tex]$18 - No initiation fee Explicit Form of Arithmetic Sequence: \[ a_n = 18 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 18x \] where \( x \) is the number of months, and \( y \) is the total cost. #### Option 2 Details: - Initiation fee: $[/tex]10
- Monthly cost: [tex]$15 Explicit Form of Arithmetic Sequence: \[ a_n = 10 + 15 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 15x + 10 \] where \( x \) is the number of months, and \( y \) is the total cost. #### Option 3 Details: - Initiation fee: $[/tex]75
- Monthly cost: [tex]$5 Explicit Form of Arithmetic Sequence: \[ a_n = 75 + 5 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 5x + 75 \] where \( x \) is the number of months, and \( y \) is the total cost. Here’s the summary in a table: \begin{tabular}{|c|c|c|} \hline Option 1 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 18 \cdot n[tex]$ & $[/tex]y = 18x[tex]$ \\ \hline Option 2 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 10 + 15 \cdot n[tex]$ & $[/tex]y = 15x + 10[tex]$ \\ \hline Option 3 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 75 + 5 \cdot n[tex]$ & $[/tex]y = 5x + 75[tex]$ \\ \hline \end{tabular} ### 2. Similarity between Explicit Form of Arithmetic Sequences and Slope-Intercept Form The explicit form of arithmetic sequences and the slope-intercept form for linear functions both consist of a starting point (initial value) and a rate of change. - In the arithmetic sequence \( a_n = a_1 + (n-1)d \): - \( a_1 \) represents the initial term. - \( d \) is the common difference between terms. - In the linear function \( y = mx + b \): - \( b \) represents the y-intercept (initial value when \( x \) is 0). - \( m \) is the slope (rate of change). Both forms identify an initial value and how much the value changes per unit (month). ### 3. Interpretation of \( y \)-Intercept and Slope for Each Option Option 1: - Y-intercept (0): No initiation fee. - Slope (18): The total cost increases by $[/tex]18 per month.
Option 2:
- Y-intercept (10): The [tex]$10 initiation fee. - Slope (15): The total cost increases by $[/tex]15 per month.
Option 3:
- Y-intercept (75): The [tex]$75 initiation fee. - Slope (5): The total cost increases by $[/tex]5 per month.
For each option, the y-intercept reflects the initial cost Karla has to pay to start the plan, and the slope indicates the ongoing monthly charge based on the plan's rate.
### 1. Arithmetic Sequence and Linear Function for Each Option
We need to derive both the arithmetic sequence in explicit form and the linear function in slope-intercept form for each subscription option.
#### Option 1
Details:
- Monthly cost: [tex]$18 - No initiation fee Explicit Form of Arithmetic Sequence: \[ a_n = 18 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 18x \] where \( x \) is the number of months, and \( y \) is the total cost. #### Option 2 Details: - Initiation fee: $[/tex]10
- Monthly cost: [tex]$15 Explicit Form of Arithmetic Sequence: \[ a_n = 10 + 15 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 15x + 10 \] where \( x \) is the number of months, and \( y \) is the total cost. #### Option 3 Details: - Initiation fee: $[/tex]75
- Monthly cost: [tex]$5 Explicit Form of Arithmetic Sequence: \[ a_n = 75 + 5 \cdot n \] where \( n \) is the number of months. Slope-Intercept Form of Linear Function: \[ y = 5x + 75 \] where \( x \) is the number of months, and \( y \) is the total cost. Here’s the summary in a table: \begin{tabular}{|c|c|c|} \hline Option 1 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 18 \cdot n[tex]$ & $[/tex]y = 18x[tex]$ \\ \hline Option 2 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 10 + 15 \cdot n[tex]$ & $[/tex]y = 15x + 10[tex]$ \\ \hline Option 3 & Explicit Form & Slope Intercept Form \\ \hline & $[/tex]a_n = 75 + 5 \cdot n[tex]$ & $[/tex]y = 5x + 75[tex]$ \\ \hline \end{tabular} ### 2. Similarity between Explicit Form of Arithmetic Sequences and Slope-Intercept Form The explicit form of arithmetic sequences and the slope-intercept form for linear functions both consist of a starting point (initial value) and a rate of change. - In the arithmetic sequence \( a_n = a_1 + (n-1)d \): - \( a_1 \) represents the initial term. - \( d \) is the common difference between terms. - In the linear function \( y = mx + b \): - \( b \) represents the y-intercept (initial value when \( x \) is 0). - \( m \) is the slope (rate of change). Both forms identify an initial value and how much the value changes per unit (month). ### 3. Interpretation of \( y \)-Intercept and Slope for Each Option Option 1: - Y-intercept (0): No initiation fee. - Slope (18): The total cost increases by $[/tex]18 per month.
Option 2:
- Y-intercept (10): The [tex]$10 initiation fee. - Slope (15): The total cost increases by $[/tex]15 per month.
Option 3:
- Y-intercept (75): The [tex]$75 initiation fee. - Slope (5): The total cost increases by $[/tex]5 per month.
For each option, the y-intercept reflects the initial cost Karla has to pay to start the plan, and the slope indicates the ongoing monthly charge based on the plan's rate.
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