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Math Help - Geometric Series & A Classic Problem

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    Geometric Series & A Classic Problem

    Can someone please put me on the right road to solving these problems? Anyone who can help, please do. It would be greatly appreciated. Please see attached. Thanks
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    Quote Originally Posted by fw_mathis View Post
    Can someone please put me on the right road to solving these problems? Anyone who can help, please do. It would be greatly appreciated. Please see attached. Thanks
    "Use the geometric sequence of numbers 1, 1/3, 1/9 , 1/27… to find the following:
    What is r, the ratio between 2 consecutive terms?"
    r = a2/a1 = (1/3)/1 = 1/3
    r = a3/a2 = (1/9)/(1/3) = (1/9)(3/1) = 3/9 = 1/3
    r = a4/a3 = (1/27)/(1/9) = 9/27 = 1/3

    "Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Carry all calculations to 6 decimals on all assignments."
    Sn = [a1*(1 -r^n)] /(1-r)
    S(10) = [1*(1 -(1/3)^10]/(1 -1/3)
    S(10) = 1.499975 -----------------------answer.

    --------------------------------------------------------------
    "CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Crane came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Crane insisted on giving the man an award for his heroism.

    So, the salesman said, “If you insist, I do not want much. Get your checkerboard and place one grain of wheat on the first square. Then place two grains of wheat on the next square. Then place four grains on the third square. Continue this until all 64 squares are covered with grains of wheat.” As he had just harvested his wheat, Mr. Crane did not consider this much of an award, but he soon realized he made a miscalculation on the amount of wheat involved."

    Comment:
    Yes, Mr. Crane soon realized that by the time he reached the 10th square, he had problem putting the 2^9 = 512 grains of wheat inside the little square. And at the 11th square, he could no longer put the 2^10 = 1024 grains of wheat on the small square.

    "How much wheat would Mr. Crane have to put on the 24th square?"
    Answer: Cannot. Impossible. At the 11th square he stopped.

    "How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?"
    Answer: A heap. Not much. How many grains of wheat could be loaded on a checker board of 24 squares? Not much.

    Calculate the amount of wheat necessary to fill the whole checkerboard (64 squares). How much wheat would the farmer need to give the salesman? Please provide the answer in either scientific notation, or calculate and show all 20 digits.
    Answer: The amount of wheat depends on the angle of repose of the wheat grains. Also, it depends on whether it is windy and on the magnitude of the force due to the wind blowing on the wheat pile. Also, it depends on whether somebody taps on the deck where the checker board is resting. Also, on whether Mr. Crane or the Salesman sneezes on the wheat pile.
    In any case, not much wheat.

    ----------------------------------------
    Now, suppose the Salesman asked Mr. Crane only some grains of wheat that doubles in quantity 24 times, starting from just one grain of wheat. You know, first--1 grain, 2nd--2 grains, 3rd--4 grains, 4th--8 grains, ....until the 24th time.

    It's geometric series. a1 = 1, r = 2, n = 24.
    S(24) = [1(1 -2^(24))]/(1-2) = 16,777,215 grains of wheat.
    Whew, how could Mr. Crane count those? The first million alone will take him about 11 and a half days, non-stop, no sleeping, no eating, no going to bathroom, if he can count 1 grain per second?

    If 64 times?
    Suppose the farmer can still count, and he has unlimited amount of grains of wheat,
    S(64) = [1(1 -2^64)]/(1-2) = 1.844674407 *10^19 grains of wheat.
    Should I ask how the farmer can even attempt to count those?
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