Let's analyze the situation to determine which recursive formula correctly represents the total amount of money in Victoria's account at the end of the [tex]\( n \)[/tex]-th year.
1. Initial Amount: Victoria has \[tex]$200 in her account at the end of the first year.
2. Deposit and Interest:
- At the beginning of each subsequent year, she deposits \$[/tex]15 into the account.
- Then, she earns 2% interest on the new balance, compounded annually.
To construct the recursive formula, let's first understand the account balance changes step-by-step:
- At the beginning of the [tex]\( n \)[/tex]-th year, she adds \[tex]$15 to the previous year's balance.
- Then, she earns 2% interest on this new balance.
Let \( a_{n-1} \) represent the total amount at the end of the \((n-1)\)-th year. At the beginning of the \( n \)-th year, the balance, after adding the deposit but before interest, is \( a_{n-1} + 15 \). Then, she earns 2% interest on this new total.
The recursive formula to express this is:
\[ a_n = 1.02(a_{n-1} + 15) \]
Now let's determine the initial condition:
- At the end of the first year, after adding the deposit and interest, she has \$[/tex]215.
Thus, the initial condition is [tex]\( a_1 = 215 \)[/tex].
Therefore, the correct recursive formula is:
[tex]\[ a_n = 1.02(a_{n-1} + 15) \quad \text{with} \quad a_1 = 215 \][/tex]
From the provided options, this corresponds to:
[tex]\[ a_n = 1.02(a_{n-1} + 15); \quad a_1 = 215 \][/tex]
So, the correct answer is:
[tex]\[ a_n = 1.02(a_{n-1} + 15); \quad a_1 = 215 \][/tex]
