To determine the correct recursive formula for Martina's savings account, let's analyze the given sequence: [tex]\(100, 105, 110.25, \ldots\)[/tex].
### Step-by-step analysis:
1. Identify the initial amount:
The initial deposit is [tex]\( \$100 \)[/tex].
2. Evaluate the first year amount:
The sequence shows that the amount at the end of the first year is [tex]\( \$105 \)[/tex].
3. Calculate the growth from the first year to the second year:
Compare the amounts between the first and second year:
[tex]\[
\text{First year amount} = 105
\][/tex]
[tex]\[
\text{Second year amount} = 110.25
\][/tex]
4. Calculate the ratio of each year:
[tex]\[
\text{Ratio from initial to first year} = \frac{105}{100} = 1.05
\][/tex]
[tex]\[
\text{Ratio from first year to second year} = \frac{110.25}{105} = 1.05
\][/tex]
We notice that the ratio is consistently [tex]\( 1.05 \)[/tex].
5. Formulate the recursive formula:
Given that each year's amount is 1.05 times the previous year's amount, the recursive formula is:
[tex]\[
f(n+1) = 1.05 \times f(n)
\][/tex]
Therefore, considering the pattern in Martina’s savings and the constant ratio of 1.05 each year, the correct recursive formula is:
[tex]\[
f(n+1) = 1.05 f(n).
\][/tex]
