# Math Help - Summation with initial conditions

1. ## Summation with initial conditions

I posted in business math, but this may be more suitable in the pre-calc forums, since I need a better understanding of an equation to use. 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!

2. Do you know the formula for the sum of a geometric series?

The series you are talking about $\{t_1,t_2,t_3,\dots\} = \{200,100,50,\dots\}$ is geometric because the ratio of any consectutive numbers is the same.

The ratio $r = \frac{200}{100}= \frac{1000}{50}= \frac{1}{2}$

with the first number $a = 200$

The some of the series can be found by $S_n =\frac{a(1-r^n)}{1-r}$

If the series is infinite then use $S_{\infty} =\frac{a}{1-r}$