Recognizing Patterns
Pre-Test Active
Martina opens a savings account with an initial deposit and makes no other deposits or withdrawals. She earns interest on her initial deposit. The total amount of money in her savings account at the end of each year is represented by the sequence shown.
$100,105,110.25, \ldots$
Which recursive formula can be used to determine the total amount of money earned in any year based on the amount earned in the previous year?
$f(n+1)=f(n)+5$
$f(n+1)=5 f(n)$
$f(n+1)=1.05 f(n)$
$f(n+1)=0.05 f(n)$
Answer
The final answer is: \(\boxed{f(n+1)=1.05 f(n)}\)
Step 1 ：Recognize the pattern in the sequence: $100,105,110.25, \ldots$
Step 2 ：Observe that the amount of money in the account increases by 5% each year. This is because the difference between 105 and 100 is 5, which is 5% of 100. Similarly, the difference between 110.25 and 105 is 5.25, which is 5% of 105.
Step 3 ：Therefore, the recursive formula that can be used to determine the total amount of money earned in any year based on the amount earned in the previous year is $f(n+1)=1.05 f(n)$
Step 4 ：The final answer is: \(\boxed{f(n+1)=1.05 f(n)}\)
