Results 1 to 2 of 2

Math Help - Mathematical Induction

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    124

    Mathematical Induction

    Prove, by mathematical induction, that

    (1^3) + (2^3) + (3^3) + ... + (n^3) = ((n(n+1))/2)^2 for all integers n≥1
    I got to here (below) and am stuck...
    (1^3) + (2^3) + (3^3) + ... + ((n+1)^3) = (((n(n+1))/2)^2) + (n+1)^3
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by bearej50 View Post
    Prove, by mathematical induction, that

    (1^3) + (2^3) + (3^3) + ... + (n^3) = ((n(n+1))/2)^2 for all integers n≥1
    I got to here (below) and am stuck...
    (1^3) + (2^3) + (3^3) + ... + ((n+1)^3) = (((n(n+1))/2)^2) + (n+1)^3
    Hi bearej50,

    P(k)

    1^3+2^3+3^3+.....+k^3=\left[\frac{k(k+1)}{2}\right]^2 ?

    P(k+1)

    1^3+2^3+3^3+.....+k^3+(k+1)^3=\left[\frac{(k+1)(k+2)}{2}\right]^2

    1^3+2^3+3^3+.....+k^3+(k+1)^3=\left[\frac{k(k+1)}{2}\right]^2+(k+1)^3

    if the first statement is true, and this must equal \left[\frac{(k+1)(k+2)}{2}\right]^2

    Proof

    \left[\frac{k(k+1)}{2}\right]^2+(k+1)^3=(k+1)^2\frac{k^2}{4}+(k+1)^2(k+1)

    =(k+1)^2\left(\frac{k^2}{4}+k+1\right)=(k+1)^2\lef  t(\frac{k^2+4k+4}{4}\right)

    =\frac{(k+1)^2(k+2)^2}{4}=\left[\frac{(k+1)(k+2)}{2}\right]^2

    hence P(k+1) is valid if P(k) is,

    hence test for an initial value N.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: June 29th 2010, 01:10 PM
  2. Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 7th 2010, 01:22 PM
  3. Mathematical Induction
    Posted in the Algebra Forum
    Replies: 9
    Last Post: July 8th 2009, 01:27 AM
  4. Mathematical Induction
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: February 17th 2009, 12:30 PM
  5. Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 30th 2007, 04:21 PM

Search Tags


/mathhelpforum @mathhelpforum