## Banking reserve and velocity of money

Just a general question from one of my econ classes. Say I deposit $100, and from that$100, the bank is allowed to loan 1/2 that amount. The next person then takes the $50 and deposits it, allowing the bank to loan 1/2 that. The$25 is deposited and so on. What is the limit?

So I understand the sum would be 100+50+25+12.5+6.25+.... and so on to $200. The question is how would I solve this in a generic sense? I have the following equation: Sum(i=2,1000, K(i-1)/2, K1=200) In other words, the initial condition (K1) is$200 so when the summation starts, it starts with \$100 and so on (a thousand times or so). Is there a better way of creating this equation without using the initial condition? That seems to prevent me from using online solving tools.

Thanks!