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Coming up with & summing a geometric series

  1. Apr 8, 2008 #1
    1. The problem statement, all variables and given/known data
    a) Two friends, Jon and Bob, are sharing a loaf of bread. Jon eats half of the loaf, then Bob eats half of what remains, then Jon eats half of what remains and so on. How much of the loaf did each of them eat?

    b)Jon is hungrier and eats 2/3 of the loaf, then Bob eats half of what remains, then Jon eats 2/3 of what remains, and Bob eats half of what remains and so on. How much of the loaf did each of them eat

    c) Now there are Jon, Bob and Ron. Jon eats half of the loaf, then Bob eats half of the remains, and Ron eats half and so on. How much did each of them eat?


    2. Relevant equations
    For geometric series:
    [tex]\sum(c*r^n)[/tex] with n starting from 0 approaching infinity = c/(1-r), |r|<1


    3. The attempt at a solution
    I made a table upto the fourth sum for all three parts, but I'm still confused on how to come up with a general series equation. Once I get the general solution, the sum should be easy i think.
    for part a, i got for Bob--[tex]\sum(1/4)^n[/tex] where n=0 going to infinity. and for Jon im doubtful i got [tex]\sum1/2^n[/tex] where n=1 going to infinity. I dont know how to start for the other parts, don't see any obvious solutions.
     
  2. jcsd
  3. Apr 8, 2008 #2
    try to set up your series so that you have it in the form you mentioned: [tex]\sum(c*r^n)[/tex]

    It is a little counter intuitive but, when n=0 your sum should be equal to the fraction of the loaf of the FIRST sharing Jon or Bob has...

    when you write [tex]\sum(1/4)^n[/tex] and n= 0 you get 1, you need to write it so that that you get his first share, which is 1/4.

    Additionally, with geometric series, it helps if you think of c as being the very first number in the series, and r as being the ratio of each term. In other words, r is what gets multiplied each time to form the next number in the series.

    If you wrote out Bob's sereis, (1/4) + (1/16) + (1/64)+... then you can immediately figure out c (its the first term, in this case (1/4)). Now you just need to figure out what is getting multiplied each time to get the next term. In other words, what can you multiply (1/4) by to get (1/16)? does this hold true for (1/16) to (1/64)? Whatever you multiplied by will be your r.
     
    Last edited: Apr 8, 2008
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