How much must a student invest initially to have $3,000 in 4 years at 4% interest compounded monthly?

To determine how much a student must invest initially to accumulate $3,000 in 4 years at an annual interest rate of 4%, with interest compounded monthly, the appropriate formula is the future value formula for compound interest:

[ A = P(1 + r/n)^{nt} ]

where:

• ( A ) is the amount of money accumulated after n years, including interest (the future value, which in this case is $3,000).

• ( P ) is the principal amount (the initial amount of money that we need to find).

• ( r ) is the annual interest rate (decimal).

• ( n ) is the number of times that interest is compounded per year.

• ( t ) is the time the money is invested for in years.

In this scenario:

• ( A = 3000 )

• ( r = 0.04 ) (4% annual interest)

• ( n = 12 ) (since the interest is compounded monthly)

• ( t = 4 )

Now, we can rearrange the formula to solve for ( P ):

[ P = \frac{A}{(1 + r/n)^{nt}} ]

Now plugging