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Math Help - Proofs by Induction

  1. #1
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    Proofs by Induction[SOLVED]

    Ok, I think I got most of the steps covered on these problems but can't seem to get the last part correct:

    sorry, image from maple since i still have not grasp on doing the html for the symbols

    i don't know if im doing the last steps correctly or not as the other problems without the summation and factorials worked fine.

    Proofs by Induction-problemir.jpg
    Last edited by hellfire127; November 30th 2010 at 10:42 AM.
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  2. #2
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    In the second last line, you write \displaystyle\sum_{i=1}^{k+2}i\cdot2^i=\sum_{i=1}^  {k+1}i\cdot2^i+k\cdot2^k, whereas the right-hand side should be \displaystyle\sum_{i=1}^{k+1}i\cdot2^i+(k+2)\cdot2  ^{k+2}.

    A style remark. In the beginning of the induction step, you write, "If p(k) is true for all integers k >= 2, then p(k+1) is true". First, it's helpful to write if this is your goal to prove or if you have already shown it. Second, this sentence can be interpreted as (\forall k\ge 2.\;p(k))\to p(k+1), which is trivial. It is better to say, "For all k >= 2, if p(k), then p(k+1)" or "p(k) implies p(k+1) for all k >= 2".
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  3. #3
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    Quote Originally Posted by emakarov View Post
    In the second last line, you write \displaystyle\sum_{i=1}^{k+2}i\cdot2^i=\sum_{i=1}^  {k+1}i\cdot2^i+k\cdot2^k, whereas the right-hand side should be \displaystyle\sum_{i=1}^{k+1}i\cdot2^i+(k+2)\cdot2  ^{k+2}.

    A style remark. In the beginning of the induction step, you write, "If p(k) is true for all integers k >= 2, then p(k+1) is true". First, it's helpful to write if this is your goal to prove or if you have already shown it. Second, this sentence can be interpreted as (\forall k\ge 2.\;p(k))\to p(k+1), which is trivial. It is better to say, "For all k >= 2, if p(k), then p(k+1)" or "p(k) implies p(k+1) for all k >= 2".
    ok i understand that part now. so the second problem should have be [(k+1)!-1]+(k+1)(k+1!). how do you simply this and the first problem to show that they equal p(k+1)?
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  4. #4
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    so the second problem should have be [(k+1)!-1]+(k+1)(k+1!). how do you simply this and the first problem to show that they equal p(k+1)?
    [(k+1)!-1]+(k+1)(k+1!) = (k + 1)! * (1 + k + 1) - 1 = (k+2)! - 1. In the first problem you similarly factor out 2^{k+2}.
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