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Math Help - quick question

  1. #1
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    quick question

    how could this one (-3)(2)^{n-1} + 2^n(-1) simplify to -2^{n+2}+3?
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  2. #2
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    Quote Originally Posted by zpwnchen View Post
    how could this one (-3)(2)^{n-1} + 2^n(-1) simplify to -2^{n+2}+3?
    It doesn't. I'm getting
    -2^{n+2}+3.2^{n-1}
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  3. #3
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    Thanks! but i do not know why it was deduced to -2^{n+2}+3 in solution manual.
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  4. #4
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    Quote Originally Posted by zpwnchen View Post
    how could this one (-3)(2)^{n-1} + 2^n(-1) simplify to -2^{n+2}+3?
    It is not true for n=2. Is it?
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  5. #5
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    it was consistently true for all n terms.

    a_n = 2a_{n-1} -3 , a_0=-1
    (-3)(2)^{n-1} + 2^n(-1) --> -2^{n+2}+3
    Last edited by zpwnchen; November 20th 2009 at 09:05 AM.
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  6. #6
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    \forall n \geq 2, \ -3 \cdot (2)^{n-1} + -2^n \equiv 0 (mod 2)
    But -2^{n+2}+3 \equiv 1 (mod 2)

    So it is obviously not true.
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  7. #7
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    Yeh! that's right!

    I was just keep checking it " -2^{n+2}+3" against " a_n = 2a_{n-1} -3 , a_0=-1" for n terms and it was true.

    So " (-3)(2)^{n-1} + 2^n(-1)" is not right! I suppose it must be something wrong in somewhere.

    Thank you!
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  8. #8
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    Quote Originally Posted by zpwnchen View Post
    Yeh! that's right!

    I was just keep checking it " -2^{n+2}+3" against " a_n = 2a_{n-1} -3 , a_0=-1" for n terms and it was true.

    So " (-3)(2)^{n-1} + 2^n(-1)" is not right! I suppose it must be something wrong in somewhere.
    For the sequence a_n = 2a_{n-1} -3 , a_0=-1 the function f(n)=-2^{n+2}+3 does give the terms of the sequence for n=1,2,\cdots.
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