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Math Help - Sequence

  1. #1
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    Sequence

    For an increasing sequence of natural numbers a_1,\ a_2,\ a_3,\ \ldots it holds that a_{(a_k)}=3k, where k is also natural number.
    Find out value of a_{100} and a_{2010}.

    I've really no idea how to solve this. Thanks in advance.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    a_n=3^n perhaps?
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  3. #3
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    Quote Originally Posted by fhactor View Post
    For an increasing sequence of natural numbers a_1,\ a_2,\ a_3,\ \ldots it holds that a_{(a_k)}=3k, where k is also natural number.
    Find out value of a_{100} and a_{2010}.

    I've really no idea how to solve this. Thanks in advance.
    [I'm sure we have had this problem here previously, but I can't locate it.]

    To start you off, let a_1=n. Then a_n = a_{a_1}=3\times1 = 3. But the sequence is increasing, and 1<n, so it follows that a_1<a_n. In other words, n<3. Therefore the only possibility is that n=2.

    Thus a_1=2 and a_2=3. Therefore a_3 = 3\times2=6, and a_6 = 3\times3=9. But if  a_3=6 and a_6 = 9 then a_4 and a_5 have to fit in between 6 and 9 (because the sequence is increasing). The only way that can happen is if a_4=7 and a_5=8.

    Continue exploring in that way, using the facts that a_{(a_k)}=3k and that the sequence is increasing. You will find that a_k is uniquely determined for each  k. After a while, a pattern should emerge, which will enable you to predict a_{100} and a_{2010}.
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  4. #4
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    Quote Originally Posted by Opalg View Post
    [I'm sure we have had this problem here previously, but I can't locate it.]

    To start you off, let a_1=n. Then a_n = a_{a_1}=3\times1 = 3. But the sequence is increasing, and 1<n, so it follows that a_1<a_n. In other words, n<3. Therefore the only possibility is that n=2.

    Thus a_1=2 and a_2=3. Therefore a_3 = 3\times2=6, and a_6 = 3\times3=9. But if  a_3=6 and a_6 = 9 then a_4 and a_5 have to fit in between 6 and 9 (because the sequence is increasing). The only way that can happen is if a_4=7 and a_5=8.

    Continue exploring in that way, using the facts that a_{(a_k)}=3k and that the sequence is increasing. You will find that a_k is uniquely determined for each  k. After a while, a pattern should emerge, which will enable you to predict a_{100} and a_{2010}.
    Cool, thanks for help, I am on it.
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  5. #5
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    I believe I got it.
    a_9=18, a_{27}=54, a_{81}=162, so a_{100} should be 181.
    a_{729}=1458, a_{1458}=2187, so a_{2010}=2187+(2010-1458) \cdot 3=3843.

    True or false?
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