1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mathematical Induction

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data

    show n3 + n < 3n for all n >= 4

    2. Relevant equations



    3. The attempt at a solution

    I.H : n3 + n < n for all n >= 4

    3(n3 + n) < 3(3n)
    then (3n+1) = 3 x 3 n
    > 3((n3) + n ) by I.H
    > (n+1)3 + (n+1)

    if we show 3(n3 + n ) - [(n+1)3 + (n+1)] > 0 by subbin in 4 which is n >= 4, does it suffice as proof?
     
    Last edited: Sep 29, 2012
  2. jcsd
  3. Sep 29, 2012 #2
    Yes it does.
     
  4. Sep 29, 2012 #3
    > 3((n3) + n ) by I.H
    I am still unsure if I got it right....

    I said in my induction hypothesis that, n3 + n < 3n
    but while im trying to prove that p(k+1) works for all k,
    I think I am using assuming this line.... which is the p(k+1) that I am trying to prove. 3(n3 + n) < 3(3n) by Induction Hypothesis.

    so technically I am using what I need to prove to prove my question... ahaha
    im just confused, can someone check my work please, thank you.
     
  5. Sep 29, 2012 #4
    [tex]3^{n+1}=3\cdot 3^n>3(n^3+n)[/tex]
    so you need to prove:
    [tex]3(n^3+n)>(n+1)^3+(n+1)[/tex]
    and then:
    [tex]3^{n+1}>(n+1)^3+(n+1)[/tex]
    Is this what you mean?
     
  6. Sep 29, 2012 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I don't see where you have proven it for n= 6.
     
  7. Sep 29, 2012 #6
    ok I think i figured out the trick.
    I expand both sides, but on the left side i make
    n3 + n3 + n3 + 3n and compare it to the right side term by term to prove that its greater for all n > 4.

    am I right with this approach?
     
  8. Sep 30, 2012 #7
    Yes, you are.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Mathematical Induction
  1. Mathematical induction (Replies: 10)

  2. Mathematical induction (Replies: 13)

  3. Mathematical induction (Replies: 2)

  4. Mathematical Induction (Replies: 3)

  5. Mathematical induction (Replies: 3)

Loading...