# If \$37,500 is invested at 6.3% for 30 years, find the future value if the interest is compounded annually,semiannually, quarterly, monthly

## 37500(1+(.063/x))^30x

where x is the number of times it is compounded per year (1 for annually, 2 for semiannually, 4 for quarterly, 12 for monthly)

## To find the future value of an investment with compound interest, we can use the formula:

Future Value = Principal * (1 + Interest Rate / Number of Compounding Periods)^(Number of Compounding Periods * Time)

Where:
- Principal is the initial investment,
- Interest Rate is the annual interest rate as a decimal,
- Number of Compounding Periods is the number of times interest is compounded per year, and
- Time is the number of years the money is invested for.

Let's calculate the future value for each type of compounding:

1. Annually compounded interest:
Number of Compounding Periods = 1
Future Value = \$37,500 * (1 + 0.063 / 1)^(1 * 30)

2. Semiannually compounded interest:
Number of Compounding Periods = 2
Future Value = \$37,500 * (1 + 0.063 / 2)^(2 * 30)

3. Quarterly compounded interest:
Number of Compounding Periods = 4
Future Value = \$37,500 * (1 + 0.063 / 4)^(4 * 30)

4. Monthly compounded interest:
Number of Compounding Periods = 12
Future Value = \$37,500 * (1 + 0.063 / 12)^(12 * 30)

Now let's calculate the future values using these formulas.

1. For annually compounded interest:
Future Value = \$37,500 * (1 + 0.063 / 1)^(1 * 30)
Future Value = \$37,500 * (1 + 0.063)^(30)
Future Value ≈ \$37,500 * (1.063)^30
Future Value ≈ \$37,500 * 4.322531
Future Value ≈ \$162,845.33

2. For semiannually compounded interest:
Future Value = \$37,500 * (1 + 0.063 / 2)^(2 * 30)
Future Value = \$37,500 * (1 + 0.0315)^(60)
Future Value ≈ \$37,500 * (1.0315)^60
Future Value ≈ \$37,500 * 3.822342
Future Value ≈ \$143,335.65

3. For quarterly compounded interest:
Future Value = \$37,500 * (1 + 0.063 / 4)^(4 * 30)
Future Value = \$37,500 * (1 + 0.01575)^(120)
Future Value ≈ \$37,500 * (1.01575)^120
Future Value ≈ \$37,500 * 3.710882
Future Value ≈ \$139,409.57

4. For monthly compounded interest:
Future Value = \$37,500 * (1 + 0.063 / 12)^(12 * 30)
Future Value = \$37,500 * (1 + 0.00525)^(360)
Future Value ≈ \$37,500 * (1.00525)^360
Future Value ≈ \$37,500 * 3.709152
Future Value ≈ \$139,478.80

Therefore, the future values for the given investment with compounding periods of annually, semiannually, quarterly, and monthly are approximately \$162,845.33, \$143,335.65, \$139,409.57, and \$139,478.80, respectively.