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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.

[tex]\[ 100, 105, 110.25, \ldots \][/tex]

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?

A. [tex]\( f(n+1) = f(n) + 5 \)[/tex]
B. [tex]\( f(n+1) = 5f(n) \)[/tex]
C. [tex]\( f(n+1) = 1.05f(n) \)[/tex]
D. [tex]\( f(n+1) = 0.05f(n) \)[/tex]



Answer :

GinnyAnswer
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]

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