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Math Help - Word problem

  1. #1
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    Word problem

    In the well-know song "the twelve days of Christmas" a person gives his or her true love k gifts on the k-th day for each of the 12 days of Christmas and repeats each gift identically on the subsequent day. Thus the twelfth day the true love receives one gift plus two gifts plus three gifts for the third day, and so on.

    a, How many gifts are received on the the twelfth day?

    b, How many gifts are received over the entire 12 days?
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  2. #2
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    Hello,

    Quote Originally Posted by gumi View Post
    In the well-know song "the twelve days of Christmas" a person gives his or her true love k gifts on the k-th day for each of the 12 days of Christmas and repeats each gift identically on the subsequent day. Thus the twelfth day the true love receives one gift plus two gifts plus three gifts for the third day, and so on.
    a, How many gifts are received on the the twelfth day?
    On the 12th day, the true love receives 1+2+3+...+12, do you agree ?

    Remember the formula from which Gauss gifted a sample to his teacher :

    1+2+\dots+n=\frac{n(n+1)}{2}

    b, How many gifts are received over the entire 12 days?
    This is more tricky..

    Let a_n be the number of presents received on the n-th day.

    From the formula above, a_n=\frac{n(n+1)}{2}.


    We want to calculate S=a_1+a_2+\dots+a_{12}=\sum_{n=1}^{12} \frac{n(n+1)}{2}

    \begin{aligned} S &=\frac 12 \sum_{n=1}^{12} n^2+n \\<br />
&=\frac 12 \sum_{n=1}^{12} n^2+\frac 12 \sum_{n=1}^{12} n \end{aligned}

    You already know what the second term is equal to..

    Now, there is another formula you should know :

    \sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}


    And normally, it's all done
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  3. #3
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    Hello, gumi!

    Are we expected to derive the necessary formulas?
    . . Or is simple addition sufficient?


    In the well-known song "the twelve days of Christmas", a person gives his or her true love
    k gifts on the k^{th} day for each of the 12 days of Christmas and repeats each gift identically
    on each subsequent day. Thus the twelfth day the true love receives one gift plus two gifts
    plus three gifts ... plus twelve gifts.

    a. How many gifts are received on the the twelfth day?

    On the 12th day: . 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 \;=\;78 gifts.


    b. How many gifts are received over the entire 12 days?
    Total: . 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 \;=\;364 gifts.

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