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Mathematical induction inequality

  1. Sep 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove; n^2 > n+1 for n = 2,3,4 by Induction

    2. Relevant equations

    3. The attempt at a solution

    p(n)= P(2) 2^2> 2+1 --> 4>3

    Induction step:
    P(n+1): (n+1)^2 > (n+1) +1
    (n+1)^2> n+2
    n^2 + 2n + 1 > n+2 | -n
    n^2 +n + 1> 2 | -1
    n^2 +n > 1

    Is this correct, and how do I go from here?
  2. jcsd
  3. Sep 20, 2013 #2
    Hint: for what n is the inequality suppose to be valid?
  4. Sep 20, 2013 #3
    For n>1. But I Still struggle to get further..
  5. Sep 20, 2013 #4


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    Homework Helper

    You're now supposed to apply your assumption that [itex]n^2>n+1[/itex] is true. If we know that [itex]n^2>n+1[/itex] then is [itex]n^2+n>1[/itex] ?
  6. Sep 20, 2013 #5


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    Staff Emeritus
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    Better said "for n= 2, n^2= 2^2= 4> 3= 2+ 1"

    You have to first say "assume that for some n, n^2> n+1".
    (Personally I prefer to use another letter, k, say, so as not to confuse it with the general n.

    No, You are asserting what you want to prove.
    Instead look at just the left side: (n+1)^2= n^2+ 2n+ 1.
    By the "induction hypothesis", n^2> n+ 1 so (n+1)^2> (n+1)+ 2n+ 1= 3n+ 2> n+ 2= (n+1)+1

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