Results 1 to 5 of 5

Math Help - Use mathematical induction to prove the truth assertions for all n≥1

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    United States
    Posts
    16

    Use mathematical induction to prove the truth assertions for all n≥1

    I need to prove: n3 + 5n is divisible by 6.

    n=1
    13+5(1)=6 which is divisible by 6 = true.

    n=2
    23+5(2)=18 which is divisible by 6 = true.

    What else do I need to prove?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,403
    Thanks
    1486
    Awards
    1

    Re: Use mathematical induction to prove the truth assertions for all n≥1

    Quote Originally Posted by rtrumpow View Post
    I need to prove: n3 + 5n is divisible by 6.
    n=1
    13+5(1)=6 which is divisible by 6 = true.
    n=2
    23+5(2)=18 which is divisible by 6 = true.
    What else do I need to prove?
    Suppose that K^3+5K is divisble by six, where K\ge 3 is a positive integer.

    Using that fact, prove that
     \begin{align*}  (K+1)^3+5(K+1)&=(K^3+3K^2+3K+1)+(5K+5)\\&=(K^3+5K)  +3K(3K+1)+6\end{align*}
    is divisble by six.

    HINT: The sum of three multiples of six is divisble by six.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,996
    Thanks
    1130

    Re: Use mathematical induction to prove the truth assertions for all n≥1

    I'm puzzled as to why you were given a problem requiring proof by induction if, as you seem to be saying, you have no idea what "proof by induction" means.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Apr 2013
    From
    Green Bay
    Posts
    68
    Thanks
    16

    Re: Use mathematical induction to prove the truth assertions for all n≥1

    Remember that steps to induction goes like this

    Let P be a statement. Show that

    P(1) is true
    If P(k) is true, then P(k+1) is true

    You already showed that P(1) is true. Now just need to show P(k+1) is true if P(k) is true. Let P(k) be true. Then k^3 + 5k is divisible by 6. Need to show that (k+1)^3 + 5(k+1) is divisible by 6. Expand.

    (k+1)^3 + 5(k+1) = k^3 + 3k^2 + 3k + 1 + 5k + 5 = (k^3 + 5k) + 3k(k+1) + 6.

    Clearly 6 is divisible by 6. By induction assumption k^3 + 5k is divisible by 6. I claim that 3k(k+1) is divisible by 6. That is because k(k+1) is even for all k. If k is even, then clearly k(k+1) is even so 3k(k+1) is divisible by 6. If k is odd, then k + 1 is even so k(k+1) is even and again 3k(k+1) is divisible by 6. The sum of terms each of which are divisible by 6 is also divisible by 6.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Use mathematical induction to prove the truth assertions for all n≥1

    I know the OP probably needed to use induction. For completeness, I'll give a proof without induction. I hope this gives more intuition as to why the statement is true.

    Lemma
    n^3+5n is divisible by 6 if and only if n^3-n is divisible by 6.

    Proof: Suppose n^3+5n=6k, then n^3-n=6(k-n). k>n because k= \frac{n(n^2+5)}{6}\geq n since n^2+5 \geq 6 for n\in \mathbb{N}. Conversely, n^3-n=6k \Rightarrow n^3+5n=6(k+n).

    Proposition
    n^3+5n is divisible by 6.

    Proof: From the lemma, the proposition holds iff n^3-n is divisible by 6. But n^3-n=n(n^2-1)=(n-1)(n)(n+1). These are three consecutive integers, and so one of them must be divisible by 3.
    Last edited by Gusbob; April 8th 2013 at 09:07 PM.
    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. Prove By Mathematical Induction
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 30th 2010, 10:22 AM
  3. Replies: 1
    Last Post: May 12th 2010, 12:07 PM
  4. using mathematical induction to prove something
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: April 30th 2008, 07:32 PM
  5. Prove by Mathematical Induction
    Posted in the Number Theory Forum
    Replies: 10
    Last Post: June 1st 2007, 06:16 PM

Search Tags


/mathhelpforum @mathhelpforum