Problem 71 - Algebra and Trigonometry

Question

A deposit of \(\$ 6000\) is made in an account that earns \(6 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=6000\left(1+\frac{0.06}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after five years. Round to the nearest cent.

Step-by-Step Solution

Answer

The balance in the account after five years would be approximately $8004.89.

Step 1: Understand the question

The question is asking to find the balance in the account after five years. The balance in the account is deployed by a mathematical sequence \(a_{n}=6000(1+\frac{0.06}{4})^{n}\), where n equals to the number of quarters.

Step 2: Conversion of Years to Quarters

Since the interest is compounded quarterly and we have to calculate after 5 years, we convert these years into quarters. Five years will count as \(5*4 = 20\) quarters.

Step 3: Substitute n with 20

In the sequence \(a_{n}=6000(1+\frac{0.06}{4})^{n}\), replace n by 20.

Step 4: Calculate

After substituting n=20 in the sequence, calculate the expression which will give us the balance after five years.

Step 5: Round to The Nearest Cent

Once the calculation is completed, round off the answer to the nearest cent as asked by the problem.