Question

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? \\(\circ\\) \\(f(n + 1) = f(n) + 5\\) \\(\circ\\) \\(f(n + 1) = 5f(n)\\) \\(\circ\\) \\(f(n + 1) = 1.05f(n)\\) \\(\circ\\) \\(f(n + 1) = 0.05f(n)\\)

Explanation:

Step1: Analyze the sequence type

The sequence is 100, 105, 110.25, ... . Let's check the ratio between consecutive terms.
First, $\frac{105}{100} = 1.05$, and $\frac{110.25}{105} = 1.05$. So it's a geometric sequence with a common ratio of 1.05.

Step2: Determine the recursive formula

For a geometric sequence, the recursive formula is $f(n + 1)=r\times f(n)$, where $r$ is the common ratio. Here, $r = 1.05$, so the formula is $f(n + 1)=1.05f(n)$.

Answer:

$f(n + 1) = 1.05f(n)$ (the third option)