How much should a family deposit at the end of every 6 month

How much should a family deposit at the end of every 6 months in order to have $4000 at the end of 3 years? The account pays 5.5% interest compounded semiannually. (Round your final answer to two decimal places.)

Solution

Let the amount needs to be deposited be A,

since it is 3 years, there will be 6 deposits

the first deposit\'s value at the end of three years = Amount * [1 + (r/100)/n]^nt = A*(1+(5.5/100)/2)^(2*3) = A[1.0275]^6

similarly,

the second deposit\'s value at the end of two and half years(since it is deposited after half year) =

Amount * [1 + (r/100)/n]^nt = A*[1 + (5.5/100)/2]^(2*2.5) = A*[1.0275]^5,

similarly adding all the deposit\'s future value,

A[1.0275]^6 + A[1.0275]^5 + A[1.0275]^4 + A[1.0275]^3 + A[1.0275]^2 + A[1.0275]

= A*[[1.0275]^6 + [1.0275]^5 + [1.0275]^4 + [1.0275]^3 +[1.0275]^2 + [1.0275]] = A*6.60470876135

=>

A*6.60470876135 = 4000

=>

A = $605.63