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Thread: difficult arithmetic series question about spool please help

  1. #1
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    difficult arithmetic series question about spool please help

    Q: The spool for a cassette tape has the circumference of 70 mm. The thickness of the tape makes each turn is 0.075mm longer than the pervious one. How many times will the spool turn playing a tape that is 54 metres long?

    I have attempted to answer it

    using the sum formula
     S_{n} = \frac{n}{2}(2a+(n-1)d)
    a = 70 d = common difference = 0.075

     S_{n} = \frac{n}{2}(140+(n-1)0.075)  = 54

     108000= 140000n + 75n^2-75n

    in the end I get n = -1866.4382or0.771523
    which is clearly wrong

    I cant seem to find the answer when I try to use the nth term

    please help
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    Re: difficult arithmetic series question about spool please help

    I just see this as

    $\displaystyle \ell = \sum C_k = \sum^n_{k=0} (70+0.075k) = 54000$

    $70(n+1) + 0.075 \dfrac{n(n+1)}{2} = 54000$

    $0.0375 n^2 + (70+0.0375)n -53930= 0$

    the positive solution is

    $n=586.1~turns$
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    Re: difficult arithmetic series question about spool please help

    Quote Originally Posted by romsek View Post
    the positive solution is
    $n=586.1~turns$
    Yer out by ~1: 587.0936477344342039884...

    a = 1st term = 70
    d = difference = .075
    t = total = 54000

    dn^2 + (2a - d)n - 2t = 0

    To the corner for ~1 minute...
    (enough time to eat 12 Easter eggs)
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    Re: difficult arithmetic series question about spool please help

    Quote Originally Posted by DenisB View Post
    Yer out by ~1: 587.0936477344342039884...
    $586 \to 53989.3 ~mm$

    $587 \to 54103.4~mm$
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    Re: difficult arithmetic series question about spool please help

    $n$ = number of turns ... (?)
    Attached Thumbnails Attached Thumbnails difficult arithmetic series question about spool please help-spool_prob.png  
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    Re: difficult arithmetic series question about spool please help

    Soooooooooooooo: whatzit me did wrong?
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    Re: difficult arithmetic series question about spool please help

    Quote Originally Posted by DenisB View Post
    Soooooooooooooo: whatzit me did wrong?
    I'm starting my sum at n=0?
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    Re: difficult arithmetic series question about spool please help

    Ahem...to the corner I go....
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