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Math Help - Summation with initial conditions

  1. #1
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    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!
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  2. #2
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    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}
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