Results 1 to 5 of 5

Math Help - Induction of inequality

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    Danmark
    Posts
    2

    Induction of inequality

    I am having a hard time getting started on this.

    n*3n<= 4n-2
    where n>6

    I have done the start step setting n = 7 and affirming that it is true.

    and I assume

    k*3k<= 4k-2

    but I cannot grasp how I should prove
    (k+1)*3(k+1)<= 4(k+1)-2

    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member DeMath's Avatar
    Joined
    Nov 2008
    From
    Moscow
    Posts
    473
    Thanks
    5

    Re: Induction of inequality

    Your inequality is very similar to the Bernoulli's inequality.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2012
    From
    Danmark
    Posts
    2

    Re: Induction of inequality

    I fail to see how.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,616
    Thanks
    1579
    Awards
    1

    Re: Induction of inequality

    Quote Originally Posted by dowant View Post
    n*3n<= 4n-2 where n>6
    I have done the start step setting n = 7 and affirming that it is true.
    and I assume
    k*3k<= 4k-2
    Start from the above.
     \begin{align*}(k+1)3^{k+1}&=(3k+3)3^k \\&\le (3k+3)(4^k-2) \end{align*}
    Last edited by Plato; September 13th 2012 at 02:05 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,528
    Thanks
    773

    Re: Induction of inequality

    This can be proved using the auxiliary fact

    3^{n+1}\le4^n for n\ge4 (*)

    It is easily proved by induction.

    \begin{align*}(n+1)\cdot3^{n+1}&=3n\cdot 3^n+3^{n+1}\\ &\le3(4^n-2)+3^{n+1}&&\text{by IH}\\ &<3\cdot 4^n-2+3^{n+1}\\ &<3\cdot4^n+4^n-2&&\text{by (*))}\\ &=4^{n+1}-2 \end{align*}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Induction: an inequality
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: June 16th 2011, 06:42 PM
  2. Induction Inequality
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 17th 2011, 01:47 PM
  3. Induction for an inequality
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 5th 2010, 10:51 PM
  4. induction inequality
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 26th 2008, 08:46 AM
  5. Induction Inequality
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 10th 2008, 08:21 PM

Search Tags


/mathhelpforum @mathhelpforum