Answer :
To solve this problem, we need to determine which recursive function rule accurately models the total amount of money in Kylie’s piggy bank at the end of each month.
Let's break it down step by step:
1. Initial Amount in the Piggy Bank:
- Kylie starts with [tex]$145. This is the amount in her piggy bank at the beginning, which we can represent as \(a_1 = 145\).
2. Monthly Addition:
- Each month, Kylie adds $[/tex]20 to her piggy bank. This means the amount of money increases by [tex]$20 every month.
3. Recursive Function Rule:
- A recursive function rule typically shows how a sequence is generated. In this case, it describes how much Kylie has in her piggy bank at the end of each month based on the previous month’s total.
4. Understanding the options:
- We need a rule that describes adding $[/tex]20 each month to the previous month's total.
- The formula [tex]\(a_n = 20 + a_{n-1}\)[/tex] means that the amount for the nth month is equal to [tex]$20 added to the amount from the previous month (\(a_{n-1}\)). This matches the condition of adding $[/tex]20 each month.
- The initial condition [tex]\(a_1 = 145\)[/tex] correctly reflects Kylie’s starting amount.
Therefore, the recursive function rule that models this situation is [tex]\(a_n = 20 + a_{n-1}\)[/tex] with [tex]\(a_1 = 145\)[/tex]. This corresponds to option 3 in the given choices.
Let's break it down step by step:
1. Initial Amount in the Piggy Bank:
- Kylie starts with [tex]$145. This is the amount in her piggy bank at the beginning, which we can represent as \(a_1 = 145\).
2. Monthly Addition:
- Each month, Kylie adds $[/tex]20 to her piggy bank. This means the amount of money increases by [tex]$20 every month.
3. Recursive Function Rule:
- A recursive function rule typically shows how a sequence is generated. In this case, it describes how much Kylie has in her piggy bank at the end of each month based on the previous month’s total.
4. Understanding the options:
- We need a rule that describes adding $[/tex]20 each month to the previous month's total.
- The formula [tex]\(a_n = 20 + a_{n-1}\)[/tex] means that the amount for the nth month is equal to [tex]$20 added to the amount from the previous month (\(a_{n-1}\)). This matches the condition of adding $[/tex]20 each month.
- The initial condition [tex]\(a_1 = 145\)[/tex] correctly reflects Kylie’s starting amount.
Therefore, the recursive function rule that models this situation is [tex]\(a_n = 20 + a_{n-1}\)[/tex] with [tex]\(a_1 = 145\)[/tex]. This corresponds to option 3 in the given choices.
