Results 1 to 6 of 6

Math Help - Mathematical induction

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    80

    Mathematical induction

    Use the principle of mathematical induction to prove the
    n
    Σ r^3 = 1/4n^2(n+1)^2.
    r=1

    I need help on this problem.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi

    You can see it works for n=1, i.e. \sum\limits_{r=1}^1r^3=\left( \frac{1(1+1)}{2}\right)^2

    Now assume it is true for some n\geq 0.

    Write:

    \sum\limits_{r=1}^{n=1}r^3
    =\sum\limits_{r=1}^nr^3+(n+1)^3
    =\left( \frac{n(n+1)}{2}\right)^2+(n+1)^3 (by induction hypothesis)
    =\left( \frac{n(n+1)}{2}\right)^2+(n+1)^2(n+1)

    Can you end?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    80
    Thank you, but I cant end till I see. Once I see it I can solve another problem. I would appreciate if ou can end it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Well the idea is to factorize by (n+1)^2 which appears in the two terms:

    \left( \frac{n(n+1)}{2}\right)^2+(n+1)^2(n+1)
    =(n+1)^2\left( \frac{n^2}{4}+(n+1)\right)
    =(n+1)^2\left( \frac{n^2+4n+4}{4}\right)
    =(n+1)^2\left( \frac{(n+2)^2}{4}\right)
    =\left( \frac{(n+1)(n+2)}{2}\right)^2

    which is what you wanted.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2009
    From
    Africa
    Posts
    641

    Smile

    Quote Originally Posted by clic-clac View Post
    Hi

    You can see it works for n=1, i.e. \sum\limits_{r=1}^1r^3=\left( \frac{1(1+1)}{2}\right)^2

    Now assume it is true for some n\geq 0.

    Write:

    \sum\limits_{r=1}^{n=1}r^3
    =\sum\limits_{r=1}^nr^3+(n+1)^3
    =\left( \frac{n(n+1)}{2}\right)^2+(n+1)^3 (by induction hypothesis)
    =\left( \frac{n(n+1)}{2}\right)^2+(n+1)^2(n+1)

    Can you end?
    i think he should assume that it is true for n\geq 1,no ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Yeah indeed it would be better according to what was done before

    By the way the case n=0 is also true, but isn't very interesting (nothing more than 0=0)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: June 29th 2010, 12:10 PM
  2. Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 7th 2010, 12:22 PM
  3. Mathematical Induction
    Posted in the Algebra Forum
    Replies: 9
    Last Post: July 8th 2009, 12:27 AM
  4. Mathematical Induction
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: February 17th 2009, 11:30 AM
  5. Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 30th 2007, 03:21 PM

Search Tags


/mathhelpforum @mathhelpforum