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Math Help - set, geometric sequence,fraction and arithmetic sequence

  1. #1
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    set, geometric sequence,fraction and arithmetic sequence

    please try to solve that questions.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by m777 View Post
    please try to solve that questions.
    Show that the set 2 \mathbb{Z} of even integers is countable.

    I presume you know the theorem that if two sets are isomorphic then they have the same cardinality?

    The function f: \mathbb{Z} \to 2\mathbb{Z}: x \mapsto 2x is an isomorphism between the two sets as you can easily show. Thus since \mathbb{Z} is countably infinite so is the set 2 \mathbb{Z}.

    (If you don't have this theorem available, note that you can put the two sets in one to one correspondence using the above function and go from there.)

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by m777 View Post
    please try to solve that questions.
    Find the 9th term of the geometric series 6, 12, 24, ...

    Since it's a geometric series we know that it is of the form:
    a_{n+1} = ra_n

    or

    a_n = a_1r^{n-1}

    So a_1 = 6

    12 = 6 \cdot r^{2-1}

    So r = 2

    The 9th term of the series is for n = 9:
    a_9 = 6 \cdot 2^8 = 1536

    -Dan
    Last edited by topsquark; November 14th 2006 at 03:53 AM.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by m777 View Post
    please try to solve that questions.
    Find the common fraction for the recurring decimal 0.7.

    We may represent the decimal by the geometric series:
    \sum_{n = 1}^{\infty} 7(10)^{-n} = 0.77777....

    The sum of the terms of a geometric series a_1r^{-n} is
    S = \frac{a_1}{r-1}

    In this case:
    S = \frac{7}{10-1} = \frac{7}{9}

    -Dan
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by m777 View Post
    please try to solve that questions.
    Which term of the arithmatic series 5, 2, -1, ... is -85?

    An arithmatic series is defined as:
    a_n = a_{n-1} + k

    or

    a_n = a_1 + (n-1)k

    We know that a_1 = 5 and 2 = 5 - (2-1)k. Thus k = -3.

    -85 = 5 + (n-1)(-3)

    -90 = -3n + 3

    -93 = -3n

    n = 31

    So the 31st term will be -85.

    -Dan
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    Hello, m777!

    Here's alternate approach to #2 . . .


    Find the common fraction for the recurring decimal 0.7777...

    We have: . N \;= \;0.77777\hdots

    Multiply by 10: . 10N \;=\;7.77777\hdots
    Subtract N: . . . . N \;= \;0.77777\hdots
    And we have: . . 9N\:=\:7\quad\Rightarrow\quad N \,=\,\frac{7}{9}

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    Quote Originally Posted by topsquark View Post
    I presume you know the theorem that if two sets are isomorphic then they have the same cardinality?
    A theorem ! You probably mean a definition.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    A theorem ! You probably mean a definition.
    It probably depends on the text you are using. My Topology book defines cardinality first, then introduces a proof that two sets with the same cardinality are necessarily isomorphic.

    Frankly your definition might be a cleaner approach. The book rather confused me when going over this.

    -Dan
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    Quote Originally Posted by topsquark View Post
    [LIST][*]It probably depends on the text you are using.[/LIST My Topology book defines cardinality first,

    -Dan
    Please show me how.
    Never seen how to define cardinality without bijections maps.
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  10. #10
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    I would like to see that topology text.
    If this is so, that text is to be avoided.
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  11. #11
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Please show me how.
    Never seen how to define cardinality without bijections maps.
    Quote Originally Posted by Plato View Post
    I would like to see that topology text.
    If this is so, that text is to be avoided.
    I owe apologies to both of you gentlemen. The word I was remembering before the bijections was "countable" not "cardinality." Yes, in my book also it defines the bijection between sets as a definition for two sets to have the same cardinality.

    -Dan
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  12. #12
    Senior Member TriKri's Avatar
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    What's a countable set and how can an infinitly big set be countable?
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  13. #13
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    Quote Originally Posted by TriKri View Post
    What's a countable set and how can an infinitly big set be countable?
    Georg Cantor developed one of the Jewels from 20th Century mathematics. The concept of sizes of sets (even infinite sets). So some infinite sets have more elements than others. Countable is any finite set, or any set which has as many elements as the number of integers. It can be shown to be the smallest type of infinite set. The Countiuum are the real numbers and they are larger. I can explain the theory but if you are not familar with the some set theory you will not follow.
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