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 must observe the relationship between consecutive terms in the sequence.
Given the sequence:
[tex]\[
100, 105, 110.25, \ldots
\][/tex]
Let's identify the relationship step-by-step:
1. The initial deposit at the end of the first year is:
[tex]\[
f(1) = 100
\][/tex]
2. At the end of the second year, the amount is:
[tex]\[
f(2) = 105
\][/tex]
Comparing this with the first year, we see:
[tex]\[
f(2) = 1.05 \times f(1)
\][/tex]
So, [tex]\(105 = 1.05 \times 100\)[/tex].
3. At the end of the third year, the amount is:
[tex]\[
f(3) = 110.25
\][/tex]
Comparing this with the second year, we see:
[tex]\[
f(3) = 1.05 \times f(2)
\][/tex]
So, [tex]\(110.25 = 1.05 \times 105\)[/tex].
Thus, the relationship between consecutive terms is that each term is 1.05 times the previous term. This can be captured in the following recursive formula:
[tex]\[
f(n+1) = 1.05 f(n)
\][/tex]
This recursive formula correctly represents the pattern in the given sequence. Therefore, the correct answer is:
[tex]\[
f(n+1) = 1.05 f(n)
\][/tex]
