Calculus question?

As soon as you start your job after graduation, you plan to start saving for retirement. You figure you will work for 40 years, and then live for another 30 years after you stop working. You want to have enough in your retirement account to be able to withdraw $3200 per month for the entire time of your retirement, but you don't plan to leave any money behind for your children. With luck, you will be able to find an investment that earns 0.55% per month.





How much money will you need 40 years from now in order to accomplish this financial objective? Remember that your account continues to earn interest, even though you are no longer making contributions (in fact you're taking money out).


$_____________





How much must you deposit each month over the next 40 years so that you will have the required amount for retirement?


$___________|||The amount of money in your account, at the end of the n-th month of your retirement, is M_n.





M_(n+1) = M_n*(1+r) - A,





where r=0.0055, and A= 3200.





The solution for this recurrence equation is





M_n = K + B (1+r)^n





By replacing into the equation, we obtain K = A/r, and with the final value A_N = 0 (N = 30*12 = 360, the expected surviving), we can calculate B = - A/[r*(1+r)^N]. Then





M_n = (A/r) * [1 - (1+r)^(n-N)]





M_0 = (A/r) [1 - (1+r)^(-N)] = $501050.





The amount accumulated while working, R_n, satisfy the equation





R_(n+1) = R_n *(1+r) + C, with R_0 = 0.





R_n = (C/r)*[(1+r)^n - 1]





R_K = M_0, where K=12*40 = 480, the number of months of working.





C = A [1 - (1+r)^(-N)] / [(1+r)^K -1] = $213.42