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Answer :
Let's solve the problem step-by-step to determine which recursive function rule models the total amount in Kylie's piggy bank at the end of each month.
1. Initial Amount:
- Kylie starts with [tex]$145 in her piggy bank. This is the starting amount, meaning the amount at the end of the first month will be $[/tex]145. Therefore, [tex]\( a_1 = 145 \)[/tex].
2. Monthly Addition:
- Each month, Kylie adds [tex]$20 to her piggy bank. This means that every month, the new total is the previous amount plus $[/tex]20.
3. Understanding the Recursive Rule:
- The recursive rule for a sequence describes how to get from one term to the next.
- Since Kylie adds [tex]$20 each month, the amount in the piggy bank after \( n \) months is the amount after \( n-1 \) months plus $[/tex]20. This can be expressed as:
[tex]\[
a_n = a_{n-1} + 20
\][/tex]
where [tex]\( a_1 = 145 \)[/tex].
Now, look at the given options to identify the correct one:
- [tex]\( a_n = 20 + a_{n-1} \)[/tex] and [tex]\( a_1 = 145 \)[/tex]: This matches our derived rule.
- Other options involve multiplication or incorrect starting amounts, which do not fit the problem description.
Therefore, the correct recursive function rule is:
[tex]\[ a_n = 20 + a_{n-1} \][/tex]
with the initial condition:
[tex]\[ a_1 = 145 \][/tex]
This choice correctly models the situation where Kylie adds [tex]$20 to her piggy bank every month, starting with an initial amount of $[/tex]145.
1. Initial Amount:
- Kylie starts with [tex]$145 in her piggy bank. This is the starting amount, meaning the amount at the end of the first month will be $[/tex]145. Therefore, [tex]\( a_1 = 145 \)[/tex].
2. Monthly Addition:
- Each month, Kylie adds [tex]$20 to her piggy bank. This means that every month, the new total is the previous amount plus $[/tex]20.
3. Understanding the Recursive Rule:
- The recursive rule for a sequence describes how to get from one term to the next.
- Since Kylie adds [tex]$20 each month, the amount in the piggy bank after \( n \) months is the amount after \( n-1 \) months plus $[/tex]20. This can be expressed as:
[tex]\[
a_n = a_{n-1} + 20
\][/tex]
where [tex]\( a_1 = 145 \)[/tex].
Now, look at the given options to identify the correct one:
- [tex]\( a_n = 20 + a_{n-1} \)[/tex] and [tex]\( a_1 = 145 \)[/tex]: This matches our derived rule.
- Other options involve multiplication or incorrect starting amounts, which do not fit the problem description.
Therefore, the correct recursive function rule is:
[tex]\[ a_n = 20 + a_{n-1} \][/tex]
with the initial condition:
[tex]\[ a_1 = 145 \][/tex]
This choice correctly models the situation where Kylie adds [tex]$20 to her piggy bank every month, starting with an initial amount of $[/tex]145.
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