Answer:
50(1.15)^n-1
Step-by-step explanation:
1. Let's consider the first three terms of g(n) to get a sense of how the function values change as n increases.
2. The first term is Luiza's account balance at the first year of the saving, which is the initial amount she deposited. We know this to be $50.
The second term is Luiza's account balance at the second year. Since the account accumulated 15% each year, it was 1.15 times the balance in the first year, which is $50*1.15=$57.50.
The third term is Luiza's account balance at the third year. Again, this is 1.15 times the balance of the year before that, which is $57.50*1.15=$66.125.
To summarize:
g(1)=50 g(2)=50*1.15 g(3)=50*1.15*1.15
We can see that each term is 1.15 times its preceding term. There is a constant ratio between consecutive terms. Therefore, this is a geometric sequence.
3. We can write an explicit formula for this geometric sequence using the form A*B^n-1. In this form, A, is the first term and B is the common ratio. What are the appropriate values for our case?
The first term is Luiza's initial deposit, which is $50.
The common ratio corresponds to the percentage of accumulated interest. Since that percentage is 15%, the common ratio is 1.15.
4. In conclusion, g is a geometric sequence.
An explicit formula for the sequence is g(n)=50*1.15^n-1
Note that this solution strategy results in this formula; however, an equally correct solution can be written in other equivalent forms as well.
