Retirement math problem (compound interest)?

Question

You begin socking away $200/month in an account that pays 6%, compounded monthly.After some months you'll have enough money in the account so that you can stop making payments and start drawing out $200/month--forever. How many months will it take?

Answer 1

Making a payment $200 per month with interest at 0.5% per month, the amount $A at the end of n months is given by:
A = 200*1.005(1.005^n - 1) / 0.005 .......(1)
If payments out start at the end of the n-th month, then to pay $200 monthly for an indefinite period requires:
A = 200 / 0.005.
Substituting this in (1):
200 / 0.005 = 200 * 1.005(1.005^n - 1) / 0.005
1.005(1.005^n - 1) = 1
1.005^n = 1 + 1.005^(-1)
n log(1.005) = log(1 + 1.005^(-1))
n = log(1 + 1.005^(-1)) / log(1.005)
= 138.48.
You therefore make 139 payments, and can start withdrawing after 139 months, when you have $40209.74 in the account: sufficient to pay $200 immediately and then $200 each subsequent month.

Answer 2

Well, obviously, if the monthly interest rate is 0.487% (which compounds to 6% a year), the principal balance needs to be $200 / 0.00487 = $41,067 to earn $200 monthly. So, if $200 is added monthly into an account earning 0.487% per month, we solve the equation 200 (1.00487^n - 1)/(0.00487) = 41067, and get n = 142 months, or almost 12 years, after a total of $28,400 has been contributed.

Answer 3

Well, if you are making 6% monthly (please tell me where you find this deal!) on your money, you would need $3400 in the bank to produce $200 in interest each month. (Technically, $3,333.33)

I've never heard of an investment that will pay you 6% interest a month on your money, so I'd be delighted if you could let me know where you are getting such a deal.

It would take roughly 17 months to have $3400 at $200 a month.