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Proof by induction

  1. Jan 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that (n+1)^4 < 4n^4 whenever n >= 3


    2. Relevant equations



    3. The attempt at a solution

    I need to prove this by induction, so i assume it is true and then prove that when n=n+1 it is also true.

    so it would become (n+2)^4 < 4(n+1)^4

    Im not sure how to continue this though.
     
  2. jcsd
  3. Jan 4, 2009 #2

    CompuChip

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    The first step is to check it for n = 3. Then assume it is true for n. I haven't tried this, but I would start by something like:
    ((n + 1) + 1)^4 = (n + 1)^4 + 4 (n + 1)^3 + ...
    Then use the induction hypothesis.

    Note that, because your question is "show that ... is STRICTLY SMALLER than ... " you can always throw away terms you know are positive. For example, for n >= 3, if you have that (n + 1)^4 < 4 n^4 then you also have that (n + 1)^4 + 14n + 3 < 4 n^4, because you are only adding terms which make the left hand side bigger so the inequality will keep holding. Watch out however, that the terms have to be positive, if (n + 1)^4 < 4 n^4 then it needn't be true that (n + 1)^4 - 20 < 4 n^4.
     
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