Results 1 to 2 of 2

Thread: Mathematical Induction

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    124

    Mathematical Induction

    Prove that 1 + 3n ≤ 4n , for every integer n ≥ 0.


    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by bearej50 View Post
    Prove that 1 + 3n ≤ 4n , for every integer n ≥ 0.


    Where did you get stuck? This one was not that difficult. I'll just give the solution. It doesn't work for zero, you can check this.


    Define a statement $\displaystyle P(n)$ by $\displaystyle P(n):~~1 + 3n \le 4n$ for all integers $\displaystyle n \ge 1$

    We show that $\displaystyle P(n)$ is true using mathematical induction.

    Since $\displaystyle 1 + 3(1) = 4 \le 4(1)$, we have that $\displaystyle P(1)$ is true.

    Assume that $\displaystyle P(n)$ is true, we show that $\displaystyle P(n + 1)$ holds also.

    Note that,

    $\displaystyle \begin{array}{rcl} 1 + 3(n + 1) & = & 1 + 3n + 3 \\ & \le & 4n + 3 \\ & \le & 4n + 4 \\ & = & 4(n + 1) \end{array}$

    So that $\displaystyle 1 + 3(n + 1) \le 4(n + 1)$, which is $\displaystyle P(n + 1)$. Thus, $\displaystyle P(n)$ holds by induction.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: Jun 29th 2010, 12:10 PM
  2. Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Apr 7th 2010, 12:22 PM
  3. Mathematical Induction
    Posted in the Algebra Forum
    Replies: 9
    Last Post: Jul 8th 2009, 12:27 AM
  4. Mathematical Induction
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Feb 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