Question

At the beginning of year 1, Carlos invests $600 at an annual compound interest rate of 4%. He makes no deposits to or withdrawals from the account. Which explicit formula can be used to find the account's balance at the beginning of year 5? What is the balance? a. A(n) = 600 • (1 + 0.04)⁽ⁿ⁻¹⁾ ; $701.92 b. A(n) = 600 • (1 + 0.04)ⁿ; $729.99 c. A(n) = 600 + (0.04 • 600)⁽ⁿ⁻¹⁾ ; $684.52 d. A(n) = 600 + (n - 1)(0.04 • 600); $696.00

Answer 1

The explicit formula for calculating the future value of Carlos's account at the beginning of year 5 is A(n) = P(1 + r)^n, which translates to A(n) = 600(1 + 0.04)^n. Using this formula for n = 4 years, the correct balance is $701.92, and the correct option is a.

Step-by-step explanation:

The question asks us to find the explicit formula for calculating the compound interest for an initial investment of $600 at an annual rate of 4% over four years, and to use that formula to calculate the account's balance at the beginning of year 5. To find the future value with compound interest, we use the formula A(n) = P(1 + r)^n, where:

• P is the principal amount (initial investment)
• r is the annual interest rate (as a decimal)
• n is the number of years the money is invested for

In this case, P = $600, r = 4% or 0.04 annually, and n = 4 years. Thus, the correct explicit formula is A(n) = 600(1 + 0.04)^n, which corresponds to option b in the question:

A(4) = 600(1 + 0.04)^4 = 600(1.04)^4

Using a calculator, we find:

A(4) = 600(1.04)^4 = 600(1.16985856) = $701.92 (rounded to two decimal places)

Therefore, correct answer is a: A(n) = 600(1 + 0.04)^(n-1) with the balance of $701.92.