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My Own Problem

  1. Jun 20, 2013 #1
    My friend was doing an analytical geometry problem and a shape appeared that I wanted to find the area of with my new knowledge of integrals. I found the area and i'm now working to find an equation for the nth area as the size of the shape changes for all integers. After doing the math I come to this sequence, (4/3), (32/3), (108/3). I stopped at the third solution because the math is a little time consuming and repetitive. Is there an equation that will represent the nth solution? Just hoping to get some help. Thanks.
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  3. Jun 20, 2013 #2


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    hi kevinnn! :smile:

    i've no idea what your area is,

    but if you divide your numbers by 4/3, you get 1, 8, 27 :wink:
  4. Jun 21, 2013 #3


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    You should understand that, just because the first few terms of a sequence have a simple relation, it is not necessary that the rest of the sequence have that relation. The classical example is the "circle region" problem: Place n points around the circumference of a circle, NOT uniformly spaced, so that when you draw all lines connecting any two of those points, no more than two such lines intersect in a single point. How many sectors does that divide the interior of the circle into?

    n= 0; 1. n= 2; 2. n= 3; 4. n= 4; 8. n= 5; 16, n= 6; 32. n= 7, 63.
    The first 7 terms are [tex]2^{n-1}[/tex] but that fails for n> 6.
  5. Jun 22, 2013 #4
    Yes i'm aware of the fact that just because it appears that a formula for the nth term in a sequence can be found it may not always exist. Do you by any chance know if it can be shown that a nth term expression for a sequence exists or doesn't exist? Possibly mathematical induction? Any other method that a first semester calculus, going into second semester, could understand? Thanks.
  6. Jun 29, 2013 #5
    Got it.

    Stat plot on a graphing calculator almost always yields insight. Once you see the points as
    (1, 4/3) (2, 32/3) (3, 108/3)
    where (x,y), things get better.

    So basically, from the graph I got, I accidentally plotted the points as their inverses where (y,x) and saw that it was definitely some power function. Quick switch to (x,y) did a power regression on the calculator (although this usually only helps you get to a rough idea, this worked great) and got a pretty obvious answer that I was too lazy to see right off the bat.


    Should've been obvious... That matches the data points perfectly for this set, try two more calculations by hand to verify that this works moderately well, although you could calculate a point off in the distance by hand, like when n=30 to see if it still works for high n.
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