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Math Help - Mathematical induction proof, help?

  1. #1
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    Mathematical induction proof, help?

    S(n): 2+7+12+...+(5n+2) = 1/2(n+1)(5n+4)

    S(1): LHS (5(1)+2) = 9
    RHS 1/2(1+1)(5(1)+4) = 9 RHS = LHS


    Assume True: S(k) = 2+7+12+...+(5k+2) = 1/2(k+1)(5k+4)

    Want to Show True: S(k+1) = 2+7+12+...+(5(k+1)+2) = 1/2((k+1)+1)(5(k+1)+4)

    Here's what I have so far.

    LHS = 2+7+12+...+ (5k+2) + (5(k+1)+2) = 5(k+1)+ 2 + 1/2(k+1)(5k+4) by induction hypothesis


    I am stuck here...if anyone can help I appreciate it.
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  2. #2
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    Quote Originally Posted by forumlurker View Post
    S(n): 2+7+12+...+(5n+2) = 1/2(n+1)(5n+4)

    S(1): LHS (5(1)+2) = 9
    RHS 1/2(1+1)(5(1)+4) = 9 RHS = LHS


    Assume True: S(k) = 2+7+12+...+(5k+2) = 1/2(k+1)(5k+4)

    Want to Show True: S(k+1) = 2+7+12+...+(5(k+1)+2) = 1/2((k+1)+1)(5(k+1)+4)

    Here's what I have so far.

    LHS = 2+7+12+...+ (5k+2) + (5(k+1)+2) = 5(k+1)+ 2 + 1/2(k+1)(5k+4) by induction hypothesis


    I am stuck here...if anyone can help I appreciate it.
    This is actually a sum of n+1 terms,
    so you should have 2+7=0.5(2)9, for n=1,
    in other words

    S_{n+1}=\frac{n+1}{2}(5n+4)

    You just need to continue on from where you were...

    You are trying to prove if 5(k+1)+2+\frac{1}{2}(k+1)(5k+4)=\frac{1}{2}(k+1+1)  (5(k+1)+4) ?

    as you are using the "fact" that the sum of (n+1) terms should be \frac{1}{2}(n+1)(5n+4)

    Here's one way to show it....

    \frac{1}{2}(k+1+1)(5(k+1)+4)=\frac{1}{2}(k+1)(5(k+  1)+4)+\frac{1}{2}(5(k+1)+4)

    =\frac{1}{2}(k+1)(5k+4)+\frac{1}{2}(k+1)5+\frac{1}  {2}(5(k+1)+4)

    =\frac{1}{2}(k+1)(5k+4)+\frac{5}{2}k+\frac{5}{2}+\  frac{5}{2}k+\frac{5}{2}+\frac{4}{2}

    =\frac{1}{2}(k+1)((5k+4)+5k+5+2 true
    Last edited by Archie Meade; February 24th 2010 at 01:28 AM. Reason: typo
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  3. #3
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    Sorry double post.
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  4. #4
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    Yes I am trying to show that these are equal.

    5(k+1)+ 2 + 1/2(k+1)(5k+4) = 1/2((k+1)+1)(5(k+1)+4)

    I understand till this point though.

    Quote Originally Posted by Archie Meade View Post
    \frac{1}{2}(k+1+1)(5(k+1)+4)=\frac{1}{2}(k+1)(5(k+  1)+4)+\frac{1}{2}(5(k+1)+4)
    My teacher has advised us to basically expand 5(k+1)+ 2 + 1/2(k+1)(5k+4) to  1/2((k+1)+1)(5(k+1)+4) to prove they are equal.

    And to stop any assumptions, this is not a homework problem, this is for me to learn. Although my instructor is helpful, he speaks quickly and is sometimes hard to understand.
    Last edited by forumlurker; February 23rd 2010 at 06:53 PM.
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  5. #5
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    Hi forumlurker,

    \frac{1}{2}(k+1\color{red}+1\color{black})(5(k+1)+  4)=\frac{1}{2}(k+1)(5(k+1)+4)+\frac{1}{2}(\color{r  ed}1\color{black})(5(k+1)+4)

    If you are trying to discover if both sides are equal,
    it's irrelevant whether you prove RHS=LHS or LHS=RHS.

    However, sometimes you run into a teacher that will stick to one way.

    5(k+1)+2+\frac{1}{2}(k+1)(5k+4)

    =\frac{1}{2}5(k+1)+\frac{1}{2}5(k+1)+\frac{1}{2}2+  \frac{1}{2}2+\frac{1}{2}(k+1)(5k+4)

    =\frac{1}{2}(5(k+1)+4)+\color{blue}\frac{1}{2}5(k+  1)+\frac{1}{2}(k+1)(5k+4)

    \color{blue}=\frac{1}{2}(k+1)(5k+5+4)\color{black}  +\frac{1}{2}(5(k+1)+4)

    =\frac{1}{2}(k+1)(5(k+1)+4)+\frac{1}{2}(5(k+1)+4)

    The first term of this sum is (k+1) times the second term,
    so we can write it as

    \frac{1}{2}(k+1+1)(5(k+1)+4) proven
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  6. #6
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    Sorry I guess this is just over my head today or something because it's not connecting. What's funny is many other induction problems aren't that difficult for me.

    Thanks for offering your time and help, not many people do that.
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  7. #7
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    The induction process is not very complex here,
    maybe it's the rearrangements of the factors you may not be used to.

    I will gladly help you understand it completely,
    tomorrow (as it's 4.40am here now!).

    Which line are you stuck on?

    You don't have to do it that way, either.
    You can simply multiply out all the terms of both sides and show they are equal!!

    \frac{1}{2}(k+2)(5k+9)=\frac{1}{2}\left(5k^2+19k+1  8\right)

    5(k+1)+2+\frac{1}{2}(k+1)(5k+4)=5k+7+\frac{1}{2}\l  eft(5k^2+9k+4\right)= \frac{1}{2}\left(10k+14+5k^2+9k+4\right)=\frac{1}{  2}\left(5k^2+19k+18\right)
    Last edited by Archie Meade; February 24th 2010 at 01:24 AM. Reason: continued
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