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Thread: Arithmetic series

  1. #1
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    Arithmetic series

    a simple one but I cant remember how to do this!!!


    Find sum of the series...

    $\displaystyle
    \sum\limits_{r = 5}^{r = 90} {(3r - 2)}
    $
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  2. #2
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    Quote Originally Posted by dankelly07 View Post
    a simple one but I cant remember how to do this!!!


    Find sum of the series...

    $\displaystyle
    \sum\limits_{r = 5}^{r = 90} {(3r - 2)}
    $
    $\displaystyle \sum_{r = 5}^{90} {(3r - 2)} $

    $\displaystyle = \sum_{r = 5}^{90}3r + \sum_{r = 5}^{90}(-2)$

    $\displaystyle = 3 \sum_{r = 5}^{90}r - 2 \sum_{r = 5}^{90}1$

    $\displaystyle = 3 \left ( \sum_{r = 1}^{90}r - \sum_{r = 1}^4r \right ) - 2 \left ( \sum_{r = 1}^{90}1 - \sum_{r = 1}^41 \right )$

    Can you take it from here?

    -Dan
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  3. #3
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    Hello,

    Quote Originally Posted by dankelly07 View Post
    a simple one but I cant remember how to do this!!!


    Find sum of the series...

    $\displaystyle
    \sum\limits_{r = 5}^{r = 90} {(3r - 2)}
    $
    Let $\displaystyle a_r=3r-2$

    $\displaystyle a_{r+1}=3(r+1)-2=(3r-2)+3=a_r+{\color{red}3}$

    Therefore :

    $\displaystyle \sum a_r$ is an arithmetic series of progression 3.

    Hmmm, can you continue ?


    --------------------

    Here is another method :

    $\displaystyle \sum_{r=5}^{90} (3r-2)=3 \sum_{r=5}^{90} r-\sum_{r=5}^{90} 2$

    You should know the general formula for $\displaystyle \sum r$.

    And for $\displaystyle \sum 2$, represent yourself adding 2, adding again, and again, and again..

    (let's see first if you can do the first one ^^)



    Edit : oh, well...topsquark spoiled it
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  4. #4
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    Hello, dankelly07!

    Another approach . . .


    Find sum of the series: .$\displaystyle \sum\limits_{r = 5}^{90} (3r - 2)$

    We have: .$\displaystyle S \:=\:13 + 16 + 19 + 22 + \hdots + 268$

    This is an arithmetic series.
    . . It has: .first term $\displaystyle a = 13$, common difference $\displaystyle d = 3$, and $\displaystyle n = 86$ terms.

    Its sum is: .$\displaystyle S \;=\;\frac{n}{2}\bigg[2a + (n-1)d\bigg] \;=\;\frac{86}{2}\bigg[2(13) + 85(3)\bigg] \;=\;12,083$

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  5. #5
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    thanks alot, all answers helped..
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