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Prove by induction

  1. Dec 9, 2011 #1
    Prove by induction that n^2 + n ≤ 2^n

    for all integers n≥5

    How i did:

    Case(1)

    Suppose that n = 5

    LHS = 5^2 +5 = 30

    RHS = 2^5 = 32

    30 ≤ 32

    Ok LHS ≤ RHS

    Case (2)

    Suppose thats true for n=p≥5. Show that its true for n = p+1

    What should i do next ? I had a memory loss here :(
     
  2. jcsd
  3. Dec 9, 2011 #2

    Fredrik

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    Just write down [itex](p+1)^2+(p+1)[/itex] and work with that expression until you see that it's [itex]\leq 2^{p+1}[/itex]. You will have to use the assumption [itex]p^2+p\leq 2^p[/itex].
     
  4. Dec 12, 2011 #3
    LHS(p+1) = (p+1)^2 + p+1 = p^2 + 2p + 1 + p + 1 = p^2 +3p + 2

    How can i continue ?
     
  5. Dec 12, 2011 #4

    Mentallic

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    Well, what's your assumption?
     
  6. Dec 12, 2011 #5
    Yes, i think so
     
  7. Dec 12, 2011 #6

    Fredrik

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    No, he asked "what's your assumption?", and you were supposed to answer "that the formula I want to prove for all n≥5 holds when n=p".
     
    Last edited: Dec 12, 2011
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