To determine the recursive formula for the sequence representing the total amount of money in Martina's savings account at the end of each year, we need to identify the pattern or relationship between the terms.
The sequence given is:
100, 105, 110.25, ...
Let's examine the first few terms:
- \( f(1) = 100 \)
- \( f(2) = 105 \)
- \( f(3) = 110.25 \)
To find the recursive relationship, let's consider the ratio between consecutive terms starting from the first term:
- \( \frac{f(2)}{f(1)} = \frac{105}{100} = 1.05 \)
- \( \frac{f(3)}{f(2)} = \frac{110.25}{105} = 1.05 \)
It appears that each term is 1.05 times the previous term. This indicates that the account balance is growing by a factor of 1.05 each year, representing a 5% increase.
Therefore, the recursive formula for the amount of money earned in the savings account based on the previous year is:
\[ f(n + 1) = 1.05f(n) \]
Thus, the correct answer is:
\[ f(n + 1) = 1.05f(n) \]
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, ...
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) = 5f(n)
f(n + 1) = 1.05f(n)
f(n + 1) = 0.05f(n)
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