Question

Luiza's savings account had $50 in its first year. Each year since then, her account accumulated interest amounting to 15% of the balance in the previous year. Let g(n) be Luiza's account balance at the n th year of the saving. g is a sequence. What kind of sequence is it?

Answer 1

`\bf \begin{array}c \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{15\% of 50}}{\left( \cfrac{15}{100} \right)50}\implies (0.15)50`

year 1................... (0.15)50

year 2................. (0.15)(0.15)50

year 3................. (0.15)(0.15)(0.15)50

year n................. 50(0.15)ⁿ

when the next term is simply obtained by multiplying the current one by some multiplier, is a geometric sequence.

Answer 2

50(1.15)^n-1

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.