# Summation with initial conditions

• Sep 23rd 2009, 12:23 PM
wildcat04
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!
• Sep 23rd 2009, 02:05 PM
pickslides
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 $S_{\infty} =\frac{a}{1-r}$