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Math Help - Elementary Math Problem

  1. #1
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    Exclamation Elementary Math Problem

    How many cubes are in the 10th collection of cubes in this sequence?

    1st- 1 cube
    2nd- 4 cubes
    3rd- 10 cubes.....
    l
    l
    l
    10th- ? cubes

    Any help appreciated!
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Tetrahedral numbers

    Hello phillyfan09
    Quote Originally Posted by phillyfan09 View Post
    How many cubes are in the 10th collection of cubes in this sequence?

    1st- 1 cube
    2nd- 4 cubes
    3rd- 10 cubes.....
    l
    l
    l
    10th- ? cubes

    Any help appreciated!
    The sequence 1, 4, 10, ... is sometimes called Tetrahedral Numbers. Here's the answer to your question: Tetrahedral number - Wikipedia, the free encyclopedia.

    Grandad
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  3. #3
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    Hello, phillyfan09!

    Where did this problem come from?
    It doesn't give enough information to allow us a good guess.

    Lucky for us, Grandad recognized the sequence.


    How many cubes are in the 10th collection of cubes in this sequence?

    . . \begin{array}{cc} <br />
\text{1st} & \text{1 cube} \\<br />
\text{2nd} & \text{4 cubes} \\<br />
\text{3rd} & \text{10 cubes} \\<br />
\vdots & \vdots \\<br />
\text{10th} & \text{? cubes} \end{array}

    Imagine a display of oranges, stacked in a triangular pyramid (tetrahedron).
    . . Each horizontal level is an equilateral triangle.

    \begin{array}{ccccc}\text{Level} & \text{array} & \text{count} & \text{Total} \\ \hline \\[-4mm] 1 & o & 1 & 1\\ 2 & \begin{array}{c}o\\ [-3mm] oo\end{array} & 3 & 4\\ 3 & \begin{array}{c}o \\ [-3mm] oo \\[-3mm] ooo\end{array} & 6 & 10 \\ 4 & \begin{array}{c} o \\[-3mm] oo\\[-3mm]ooo\\[-3mm]oooo\end{array} & 10 & 20 \\\vdots & \vdots & \vdots & \vdots \end{array}


    And we want the 10th Total on this list.


    You can crank out the numbers yourself, or you can use this formula:

    . . T(n) \;=\;\frac{n(n+1)(n+2)}{6}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    By the way, this formula answers the question:

    How many gifts did My True Love Give To Me
    . . during the Twelve Days of Christmas?

    . . \begin{array}{cccc}\text{Day} & \text{Gifts} & \text{Number} \\ \hline 1 & 1 & 1 \\ 2 &1+2 & 3 \\ 3 & 1+2+3 & 6 \\ 4 & 1+2+3+4 & 10 \\ \vdots & \vdots & \vdots \\ 12 & 1+2+\hdots+12 & 78 \\ \hline & \text{Total} & {\color{blue}364}\end{array}

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  4. #4
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    Problem came from a college level elementary course...I believe professor made it up.

    Thanks!
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