finance question

Question

Assume that you contribute $250 per month to a retirement plan for 25 years. Then you are able to increase the contribution to $500 per month for another 25 years. Given a 9.0 percent interest rate, what is the value of your retirement plan after the 50 years? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

Answer 1

so we are going to do 2 annuity questions and also a future value formula within that annuity question:

So we need to find the first 25 year annuity, then the future value of that annuity in another 25 years, then add the other 25 year annuity see below:

FV= 1.09²⁵(250((1+.09/12)^12*25-1)/(.09/12))+(500((1+.09/12)^12*25-1)/(.09/12)) = $2,977,442.19

to further explain, after 25 years the first annuity is "done" being contributed to, but will continue to earn interest for another 25 years which is where the 1.09²⁵ comes from being multiplied by the first annuity; then you have the second annuity which "starts" on the 26th year and goes for another 25 years, that one does not need to be future valued further because it goes until the end of the 50 years