We appreciate your visit to Kylie starts with tex 145 tex in her piggy bank Each month she adds tex 20 tex Which recursive function rule models the total amount. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve the problem of modeling the total amount in Kylie's piggy bank at the end of each month using a recursive function rule, let's walk through the process step-by-step:
1. Understand the initial condition:
- Kylie starts with [tex]$145 in her piggy bank. This means that at the start, before any additional money is added, her total is $[/tex]145. This is our base value, or [tex]$a_1 = 145$[/tex].
2. Determine the pattern of addition:
- Each month, Kylie adds [tex]$20 to her piggy bank. Therefore, every month, the new total is the previous month's total plus $[/tex]20.
3. Setup the recursive formula:
- A recursive formula expresses the next term in a sequence as a function of the previous term(s).
- Based on Kylie's saving pattern, if [tex]$a_{n-1}$[/tex] is the amount in the piggy bank after [tex]$(n-1)$[/tex] months, then [tex]$a_n$[/tex], the amount after [tex]$n$[/tex] months, would be the previous amount plus [tex]$20. This gives us: $[/tex]a_n = 20 + a_{n-1}[tex]$.
4. Summarize the recursive rule:
- With the initial amount (base case) of $[/tex]a_1 = 145[tex]$, and the rule for each subsequent month being $[/tex]a_n = 20 + a_{n-1}$, we can model the situation using the recursive function:
- [tex]\( a_1 = 145 \)[/tex]
- [tex]\( a_n = 20 + a_{n-1} \)[/tex]
Thus, the correct recursive function rule to model the total amount in Kylie's piggy bank at the end of each month is [tex]\( a_n = 20 + a_{n-1} \)[/tex] with [tex]\( a_1 = 145 \)[/tex].
1. Understand the initial condition:
- Kylie starts with [tex]$145 in her piggy bank. This means that at the start, before any additional money is added, her total is $[/tex]145. This is our base value, or [tex]$a_1 = 145$[/tex].
2. Determine the pattern of addition:
- Each month, Kylie adds [tex]$20 to her piggy bank. Therefore, every month, the new total is the previous month's total plus $[/tex]20.
3. Setup the recursive formula:
- A recursive formula expresses the next term in a sequence as a function of the previous term(s).
- Based on Kylie's saving pattern, if [tex]$a_{n-1}$[/tex] is the amount in the piggy bank after [tex]$(n-1)$[/tex] months, then [tex]$a_n$[/tex], the amount after [tex]$n$[/tex] months, would be the previous amount plus [tex]$20. This gives us: $[/tex]a_n = 20 + a_{n-1}[tex]$.
4. Summarize the recursive rule:
- With the initial amount (base case) of $[/tex]a_1 = 145[tex]$, and the rule for each subsequent month being $[/tex]a_n = 20 + a_{n-1}$, we can model the situation using the recursive function:
- [tex]\( a_1 = 145 \)[/tex]
- [tex]\( a_n = 20 + a_{n-1} \)[/tex]
Thus, the correct recursive function rule to model the total amount in Kylie's piggy bank at the end of each month is [tex]\( a_n = 20 + a_{n-1} \)[/tex] with [tex]\( a_1 = 145 \)[/tex].
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