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!

Prove by induction that r(r-1)(r+1) is an even integer

  1. Aug 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove by induction, that when r(r-1)(r+1) is an even integer when r=2,3,4....

    2. Relevant equations
    Prove by induction

    3. The attempt at a solution
    I began with the base case r=2, leading 6.
    Then I proceed with r=3, leading 24.

    Now if r=k is true, then k(k-1)(k+1) is also true.
    If r=k+1, then (k+1)(k)(k+2)
    But now I'm stuck at this point - how do I proceed with this?
     
  2. jcsd
  3. Aug 13, 2016 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    Write k(k-1)(k+1)=2F where F is an integer.
     
  4. Aug 13, 2016 #3
    k(k-1)(k+1) = 2F
    k^3 - k=2F

    k(k+1)(k+2) = k^3 +3k^2 +2k
    = k^3 - k +3k^2 +3k
    =2F+3k(k+1)
     
  5. Aug 13, 2016 #4

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    You can still substitute k(k+1) using k(k-1)(k+1)=2F.
     
  6. Aug 13, 2016 #5
    2F+3(2F/(k-1) ?
     
  7. Aug 13, 2016 #6

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    Yes. Now you should see why it is an even integer.
     
  8. Aug 13, 2016 #7
    Ah I see already - since 2F= 2F+6F/(k-1), when k is not equal to 1, it is divisible by 2.
     
  9. Aug 13, 2016 #8

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    I don't know if that's a typo or on purpose, but the LHS and RHS cannot be equal. Moreover, apart from being divisible by two you should also be convinced that 2F+6F/(k-1) is indeed an integer.
     
  10. Aug 13, 2016 #9
    Ah sorry about that.
     
  11. Aug 13, 2016 #10

    Mark44

    Staff: Mentor

    Somewhat OT, but in fact, (r - 1)r(r + 1) is divisible by 6, for r = 1, 2, 3, ...
     
  12. Aug 14, 2016 #11
    Looks like I managed to work them out. Please mark this thread as solved. ;)
     
  13. Sep 2, 2016 #12

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You know, induction isn't really needed here. For any ##r##, either ##r## or ##r+1## is even. Multiplying any integer by an even number yields an even number.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prove by induction that r(r-1)(r+1) is an even integer
  1. Sum of r(r+1) (Replies: 10)

Loading...