Simple answer: $10,788.61
But if you want the long answer, explained as simply as I possibly could, here it is:
First, start off by finding how much he has to pay the bank total. If it is ten dollars the first month, and it doubles every month, we get the sequence 10+20+40+80... This sequence can also be written as 10•20+10•21+10•22+10•23... or 10•(2)x-1, where x is the number of months he has to pay. If we plug in 12 months for x, we get 10•211, which simplifies to 20,480 (For Europeans, the comma is just a way to mark the thousand place, not a decimal). Now that we know how much the businessman has to pay, we have to figure out how much he can take out in the first place. 6% interest compounded monthly is a fancy way to say that every month, the businessman's debt goes up by 6% of the month before. For example, if the businessman paid $100 the first month, then the second he would pay $106, because 100+0.06•100=106. Another way of writing that is 1.06•100. This tells us that $20,480 is the twelfth month, so it would be 1.06•x11, where x11 is the total payment of the eleventh month. x11, however, is just 1.06•x10. After doing that for every month, we get 1.0611•x1=x12. But we know x12=20,480, so we can substitute that into the equation. The final equation we get is 1.0611•x1=20,480. Well, we can simplify 1.0611 to be about 1.8983, and we can plug that back into the equation, which gives us 1.8983•x1=20,480. We can divide each side by 1.8983, which gives us the final answer of x=10,788.61, which means that this businessman can take out any amount of money less than or equal to $10,788.61
