Question

Use the appropriate compound interest formula to compute the balance in the account after the stated period of time ​$4,000 is invested for 17 years with an APR of 3​% and monthly compounding.

Answer 1

After 17 years of investing $4,000 with an APR of 3% and monthly compounding, the balance in the account would be approximately $5,809.36.

Step-by-step explanation:

The compound interest formula is given by:

`\[ A = P \left(1 + (r)/(n)\right)^(nt) \]`

where:

`- \( A \)` is the future value of the investment/loan, including interest.

`- \( P \)` is the principal amount (initial investment).

`- \( r \)` is the annual interest rate (decimal).

`- \( n \)` is the number of times that interest is compounded per unit
`\( t \).`

`- \( t \)` is the time the money is invested/borrowed for in years.

In this case,
`\( P = $4,000 \), \( r = 0.03 \) (3% as a decimal), \( n = 12 \) (monthly compounding), and \( t = 17 \) years.`)

Plugging these values into the formula:

`\[ A = 4000 \left(1 + (0.03)/(12)\right)^(12 * 17) \]`

After performing the calculation, the result is approximately $5,809.36.

This formula accounts for the compounding effect, where interest is calculated not only on the initial principal but also on the accumulated interest.

In this scenario, the monthly compounding boosts the overall return compared to simple interest.

This is a common formula used in finance to determine the future value of an investment given certain parameters.

Answer 2

The formula for compound interest is
`A = P(1 + r/n)^{(nt)`, where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

Step-by-step explanation:

Understanding the Formula:

The formula for compound interest is
`A = P(1 + r/n)^{(nt)`, where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time the money is invested for in years.

Calculating the Balance:

Substituting the given values into the formula, A = 4000 * (1 + 0.03/12 ⁽¹²ˣ¹⁷⁾, the calculation will yield the balance after 17 years for a $4,000 investment at an annual interest rate of 3%, compounded monthly.

Plugging the values into the formula gives us

A = 4000 * (1 + 0.0025)²⁰⁴, simplifying it gives A = 4000 * (1.0025)²⁰⁴, resulting in the balance in the account after 17 years with monthly compounding at approximately $6,048.98.

Compound interest, when calculated using this formula, illustrates the effect of compounding frequency on the growth of an investment over time, showcasing how the interest is added to the principal amount more frequently, leading to higher overall returns.