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Math Help - Using Triangular Numbers in Advanced math

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    Using Triangular Numbers in Advanced math

    I am trying to do some research on triangular numbers. I have read many articles and books that talk about the many properties that triangular numbers have. However, I am trying to find ways to use them in advanced math or in real-world problems. I know they can be useful in some basic combinatorics but I was wondering if anyone could tell me a few other ways. Thanks.
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    Re: Using Triangular Numbers in Advanced math

    Hello, zebra2147!


    Here's an interesting fact that you may not have seen . . .


    . . . . . . . \left(\sum^n_{k=1}k\right)^2 \quad = \quad \sum^n_{k=1}k^3

    (1 + 2 + 3 + \hdots + n)^2 \:=\:1^3 + 2^3 + 3^3 + \hdots n^3

    The square of the n^{th} triangular number is equal to the sum of the first n cubes.


    Example:

    . . \begin{array}{cccc}(1+2+3+4+5)^2 &=& 225 \\ \\[-3mm] 1^3+2^3+3^3+4^3+5^3 &=& 225 \end{array}
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    Re: Using Triangular Numbers in Advanced math

    That is very interesting...I'll have to see if I can figure out why that is true. Here is something else interesting that I am investigating:

    Let tn be the nth triangular number. Then,

    t10=t5+5=55
    t100=t50+50=5050
    t1000=t500+500=500500
    t10000=t5000+5000=50005000
    ... and so on
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    Re: Using Triangular Numbers in Advanced math

    Hello again, zebra2147!


    The Triangular Numbers are evident in "The Twelve Days of Christmas".

    How many gifts did my true love give to me on the seventh day?
    . . \text{Answer: }\:T_7 \:=\:\frac{7\cdot8}{2} \:=\:28

    That may not be very exciting, but . . .


    How many gifts did my true love give to me during the entire Twelve Days?

    . . \text{Answer: }\:T_1 + T_2 + T_3 + \cdots + T_{12}

    We want the sum of the first 12 Triangular Numbers.

    Is there a formula for this? .Yes!

    The sum of the first n Triangular Numbers is: . \frac{n(n+1)(n+2)}{6}


    . . . \begin{array}{cccccccccc} &&&&&&&& \circ \\[-2mm]  &&&&&& \circ && \circ\circ \\[-2mm] &&&& \circ && \circ\circ && \circ\circ\circ \\[-2mm] && \circ && \circ\circ && \circ\circ\circ && \circ\circ\circ\circ \\[-2mm] \circ && \circ\circ && \circ\circ\circ && \circ\circ\circ\circ && \circ\circ\circ\circ\circ\\ 1 && 3 && 6 && 10 && 15\end{array}

    If we "stack" these triangles, we form tetrahedrons.

    We can restate the above formula.

    \text{The }n^{th}\text{ Tetrahedral Number is: }\:\frac{n(n+1)(n+2)}{6}
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    Re: Using Triangular Numbers in Advanced math

    Quote Originally Posted by zebra2147 View Post
    I am trying to do some research on triangular numbers. I have read many articles and books that talk about the many properties that triangular numbers have. However, I am trying to find ways to use them in advanced math or in real-world problems. I know they can be useful in some basic combinatorics but I was wondering if anyone could tell me a few other ways. Thanks.
    Here is a favorite of mine.
    Consider the sequence 1223334444555555\cdots.
    Do you see the pattern in that sequence? One 1, two 2's, three 3's, etc.
    We can use triangular numbers numbers to produce that sequence. HOW?
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