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Math Help - Confusing Sequence with Absolute value

  1. #1
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    Thumbs up Confusing Sequence with Absolute value

    Hi Forum!

    I'm having difficulties with absolute value numbers.

    This sequence is kind of strange, take a look:

    Let a be a Real number such that 1<a<2, the sequence {an} is defined by

    a1=a and an+1= |an|-1 such that {n=1,2,3,4,...}
    And granted that Sn=a1+a2+a3+...+an.

    --Find a4, a5, a6 and a7

    Ok, no problem with this one
    a2=|a1|-1 a2+1=|a1| a1=-a2-1 a2= -a+1
    a3=|a2|-1 a3+1=|-a+1| -a3-1=-a+1 a3= a-2
    a4=|a3|-1 a4+1=|a-2| -a4-1=a-2 a4= -a+1
    ...
    So, there's an alternate sequence between odd numbered and even numbered terms.
    a4=-a+1 a5=a-2
    a6=-a+1 a6=a-2

    For the problematic part:
    Find S2,S4 and S6


    S2=a1+a2
    S2=a+(-a+1) = 1

    But real answer is 2a-1
    So, what's wrong Forum?
    I really don't get what is wrong on this one.

    If someone has some tips for the absolute value part, I would humbly accept it.
    Thanks
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  2. #2
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    a_2=|a_1|-1=|a|-1 =a-1

    This follows because a is between 1 < a < 2

    a_1+a_2=a+(a-1)=2a-1
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  3. #3
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    Thumbs up

    Hi TheEmptySet!
    Thanks for the reply!

    Things like 1 < a < 2 still confuse me a bit, even more when combined with absolute values.
    So, if 1 < a < 2, how come a_4=-a+1 ?

    I'm learning!!! I'm learning!!!
    hahaha I'm loving it.

    But still, this is kind of confusing, don't you think?
    Thanks again!
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by Zellator View Post
    Hi TheEmptySet!
    Thanks for the reply!

    Things like 1 < a < 2 still confuse me a bit, even more when combined with absolute values.
    So, if 1 < a < 2, how come a_4=-a+1 ?

    I'm learning!!! I'm learning!!!
    hahaha I'm loving it.

    But still, this is kind of confusing, don't you think?
    Thanks again!
    You just need to write out the terms!

    we know that

    a_3=|a_2|-1

    but we know what a_2 is this gives

    a_3=|a-1|-1=(a-1)-1=a-2

    again this follows from 1 < a < 2

    but now when you find the next term we get

    a_4=|a_3|-1=|a-2|-1

    but

    a-2 < 0 \implies |a-2|=2-a

    This gives


    a_4=|a_3|-1=|a-2|-1=(2-a)-1=-a+1
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  5. #5
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    Oh, I thought there was a pattern starting in a_2.
    This relation between absolute values and 1<a<2 are more intimate than I thought.
    Thank you for your help.
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