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Math Help - Geometry Pattern Formula

  1. #1
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    Geometry Pattern Formula

    Hello All, thank you for your help.

    I am at a complete loss in this problem:

    Attatched are the triangular numbers used to do the problem, and a table clarifying a few things.

    The "n" row on the table states what n equals.
    The "nth Triangular Number" row states how many dots there are when n=a certain number. (So when n=1, there is one dot.)

    I am to find a formula for the nth triangular number.

    The book says that each triangular number could be thought of as half the area of a rectangle whose width is the same as the triangle number, and it's height is n+1. So if n=2, the box is thought of as 2 units wide and 3 units tall, for a total of 6 units.

    It also says that the formula for this problem is:
    S-1/2(Width)(Length)=1/2(n)(n+1)

    I have no idea how one would arrive at such an explanation for a pattern of said triangular numbers, nor how one would arrive at such a formula.

    Any help on the correct way to solve a formula such as this and how to find the formula would be most appreciated.
    Thanks a lot,
    Rocker1414
    Attached Thumbnails Attached Thumbnails Geometry Pattern Formula-triangular-number.bmp  
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  2. #2
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    sequence of triangular numbers ...

    1, 3, 6, 10, 15, ...

    double them ...

    2, 6, 12, 20, 30, ...

    factor each term ...

    1*2, 2*3, 3*4, 4*5, 5*6, ... , n(n+1)

    undouble (halve) ...

    1, 3, 6, 10, 15, ... , n(n+1)/2
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  3. #3
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    Hello, Rocker1414!



    Attatched are the triangular numbers used to do the problem, and a table clarifying a few things.

    The n row on the table states what n equals.
    The T_n row states how many dots there are when n = a certain number.

    I am to find a formula for the n^{\text{th}} triangular number.

    \begin{array}{c|cccc} n &\quad 1 & \quad2 & \quad3 & \quad4 \\ \hline<br />
& & & & \quad\bullet \\<br />
& & & \quad\bullet & \quad\bullet\:\bullet \\<br />
& & \quad\bullet & \quad\bullet\:\bullet & \quad\bullet\:\bullet\:\bullet \\<br /> <br />
& \quad\bullet & \quad\bullet\:\bullet & \quad\bullet\:\bullet\:\bullet & \quad\bullet\:\bullet\:\bullet\:\bullet \\ \hline <br /> <br />
T_n & \quad1 & \quad3 & \quad6 & \quad10<br />
\end{array}
    That hint goes like . . .


    Suppose we want to know T_5 without counting them.


    We have: . \begin{array}{c} \bullet \\ \bullet\;\bullet \\ \bullet\;\bullet\;\bullet \\ \bullet\;\bullet\;\bullet\;\bullet \\ \bullet\;\bullet\;\bullet\;\bullet\;\bullet\end{ar  ray}


    Left-justify the dots: . \begin{array}{cccccc} \bullet & & & &  \\<br />
\bullet & \bullet & & & \\<br />
\bullet & \bullet & \bullet & & \\<br />
\bullet & \bullet & \bullet & \bullet & \\<br />
\bullet & \bullet & \bullet & \bullet & \bullet\end{array}


    Add a mirror-image copy of the triangle: . \begin{array}{cccccc}\bullet& \circ&\circ&\circ&\circ&\circ \\<br />
\bullet&\bullet&\circ&\circ&\circ&\circ \\<br />
\bullet &\bullet& \bullet & \circ & \circ & \circ \\<br />
\bullet & \bullet&\bullet&\bullet&\circ&\circ \\<br />
\bullet & \bullet & \bullet & \bullet & \bullet & \circ\end{array}


    We have a 5-by-6 rectangle with a total of: . 5 \times 6 \:=\:30 dots.

    The triangle contains half that many dots.

    Therefore: . T_5 \;=\;\frac{5\cdot6}{2} \;=\; 15


    In general: . \boxed{T_n \;=\;\frac{n(n+1)}{2}}

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