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Math Help - Functions and Relations

  1. #1
    Junior Member VDestinV's Avatar
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    Exclamation Functions and Relations

    In an old story, a king asked a peasant to come and work for him. They both agreed that the job to be done could be completed in a month (30 days). The peasant asked the king if he could be paid in the following way: "On the first day that I work, I would like to be paid one penny. Each day that I work, I would like you to double my income." The king thought this was a great deal and happily agreed.

    Analyze the deal that the king and the peasant made to determine whether or not the peasant made a good deal with the king.

    Now I know that you use the formula tn=ar^n-1 but im not exactly sure how.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by VDestinV View Post
    In an old story, a king asked a peasant to come and work for him. They both agreed that the job to be done could be completed in a month (30 days). The peasant asked the king if he could be paid in the following way: "On the first day that I work, I would like to be paid one penny. Each day that I work, I would like you to double my income." The king thought this was a great deal and happily agreed.

    Analyze the deal that the king and the peasant made to determine whether or not the peasant made a good deal with the king.

    Now I know that you use the formula tn=ar^n-1 but im not exactly sure how.
    Lets write this out so...

    1(2)^0+1(2)^1+1(2)^2+...1(2)^{30}=\sum_{n=1}^{30}2  ^{n-1}

    The sum of the series will be

    S_{30}=a_1\frac{1-r^n}{1-r}=1\cdot \frac{1-2^{30}}{1-2}=2^{30}-1=1073741824 \mbox{cents}

    or in dollars

    $ 10,737,418.24 a little over ten million

    The king made a bad choice
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Actually I'm wrong that is for 31 days of labor.

    The sum should only go to 29 and should be

    536870912 cents or

    $ 5,368,709.12 about 5 million dollars.

    Good luck.
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