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

  1. Mar 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove by induction

    1^2 + 2^2 + ... + (n-1)^2 < (n^3)/3

    (This is the problem on page 33 of Apostol's book)



    2. Relevant equations



    3. The attempt at a solution

    A(k) = 1^2 + 2^2 + ... + (k-1)^2 < (k^3)/3

    A(k+1) = 1^2 + 2^2 + ... + k^2 < (k+1)^3/3


    Start with A(k) and add k^2 to both sides. This gives the inequality

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2


    To obtain A(k+1) as a consequence of this, it suffices to show that

    k^3/3 + k^2 < (k+1)^3 /3




    Now (k+1)^3 /3 = k^3/3 + k^2 +k +1/3 which is greater than k^3/3 + k^2


    Which proves the inequality. However I dont understand why does

    k^3/3 + k^2 < (k+1)^3 /3 suffice to prove the inequality

    Can anyone explain in detail why does that inequality proves it?
     
  2. jcsd
  3. Mar 13, 2010 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    They're using the transitive property of <. You know

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2.

    If you can show (k^3)/3 + k^2 < (k+1)^3/3, then you'll have

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2 < (k+1)^3/3

     
  4. Mar 13, 2010 #3
    Thanks a lot, I really appreciate it. :) I have been struggling with this for quite some time.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove by Induction
  1. Proving by induction (Replies: 1)

  2. Prove by induction (Replies: 5)

  3. Prove by Induction (Replies: 17)

Loading...