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Prove this inequality with binomial

  1. Nov 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that

    [itex]\sum_{k=0}^n {3k\choose k}\ge \frac{5^n-1}{4}[/itex]


    2. Relevant equations

    [itex]{3k\choose k}= \frac{(3k)!}{k!(2k)!}[/itex]


    3. The attempt at a solution

    I tried using the induction principle, but...

    Here my attempt:

    For [itex]n=0[/itex] 1>0 ok

    Suppose that is true for [itex]n[/itex], i.e.:

    [itex]\sum_{k=0}^n {3k\choose k}\ge \frac{5^n-1}{4}[/itex]

    Now:

    [itex]\sum_{k=0}^{n+1} {3k\choose k}= \sum_{k=0}^{n} {3k\choose k}+ {3(n+1)\choose (n+1)}\ge \frac{5^n-1}{4}+{3(n+1)\choose (n+1)}[/itex]

    But now I don't know what to do, maybe it is not the correct way to show this... I need your help
     
  2. jcsd
  3. Nov 19, 2011 #2
    Please, can someone help me? I think that it exists a different way to prove this inequality, but I don't know how to proceed :(
     
  4. Nov 19, 2011 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    What you actually need to show is that C(3(n+1),(n+1))>=(5^(n+1)-1)/4-(5^n-1)/4. Do you see why? You can definitely simplify the right side a lot. Now look at both sides for small values of n. Can you see how to continue? Think about it. I haven't much beyond this. Help me!
     
    Last edited: Nov 19, 2011
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