Answer :
To determine the recursive function rule that models the total amount in Kylie's piggy bank at the end of each month, let's break down the problem:
1. Starting Amount: Kylie starts with [tex]$145 in her piggy bank. This means that at the beginning, before adding anything, her balance is $[/tex]145. So, for the first month, we have [tex]\( a_1 = 145 \)[/tex].
2. Monthly Addition: Each month, Kylie adds [tex]$20 to her piggy bank. This means that to find the total amount in the piggy bank at the end of any month \( n \), you take the amount from the previous month and add $[/tex]20 to it.
3. Recursive Rule: A recursive function relates the current month's total to the previous month's total. Therefore, each month, the total amount becomes [tex]\( a_n = a_{n-1} + 20 \)[/tex]. This rule shows that each month's balance is the previous month's balance plus [tex]$20 extra.
Combining these statements, we interpret the recursive rule as:
- \( a_1 = 145 \), which signifies that Kylie starts with $[/tex]145 in the first month.
- [tex]\( a_n = a_{n-1} + 20 \)[/tex], which represents that every month $20 is added to the previous month's balance.
Based on this analysis, the correct rule among the options provided is:
[tex]\[ a_n = 20 + a_{n-1} \ \text{and} \ a_1 = 145 \][/tex]
1. Starting Amount: Kylie starts with [tex]$145 in her piggy bank. This means that at the beginning, before adding anything, her balance is $[/tex]145. So, for the first month, we have [tex]\( a_1 = 145 \)[/tex].
2. Monthly Addition: Each month, Kylie adds [tex]$20 to her piggy bank. This means that to find the total amount in the piggy bank at the end of any month \( n \), you take the amount from the previous month and add $[/tex]20 to it.
3. Recursive Rule: A recursive function relates the current month's total to the previous month's total. Therefore, each month, the total amount becomes [tex]\( a_n = a_{n-1} + 20 \)[/tex]. This rule shows that each month's balance is the previous month's balance plus [tex]$20 extra.
Combining these statements, we interpret the recursive rule as:
- \( a_1 = 145 \), which signifies that Kylie starts with $[/tex]145 in the first month.
- [tex]\( a_n = a_{n-1} + 20 \)[/tex], which represents that every month $20 is added to the previous month's balance.
Based on this analysis, the correct rule among the options provided is:
[tex]\[ a_n = 20 + a_{n-1} \ \text{and} \ a_1 = 145 \][/tex]
