Results 1 to 6 of 6
Like Tree7Thanks
  • 2 Post By Plato
  • 3 Post By Plato
  • 2 Post By mcolula

Math Help - Mathematical Induction with summations

  1. #1
    Newbie
    Joined
    May 2013
    From
    Mexico
    Posts
    8
    Thanks
    2

    Mathematical Induction with summations

    Hi, i've been solving some excersices but I am stuck with this one.


    Proof by Mathematical Induction the following expresion

    Mathematical Induction with summations-codecogseqn.gif

    Well the first thing I did was rewrite the expesion as this:

    Mathematical Induction with summations-codecogseqn4.gif

    then i proof this is true for p(1)

    Mathematical Induction with summations-codecogseqn3.gif

    then I assume this true for p(k)

    Mathematical Induction with summations-codecogseqn5.gif

    p(k+1)

    Mathematical Induction with summations-codecogseqn6.gif

    then

    Mathematical Induction with summations-codecogseqn7.gif

    replacing Sk

    Mathematical Induction with summations-codecogseqn8.gif

    and then i did the sum

    Mathematical Induction with summations-codecogseqn9.gif


    after this step I don't really know what to do for symplify the expression, so if can help it would be great and if i made a mistake in previous steps please make know. thanks
    Attached Thumbnails Attached Thumbnails Mathematical Induction with summations-codecogseqn2.gif  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,792
    Thanks
    1687
    Awards
    1

    Re: Mathematical Induction with summations

    Your common denominator is incorrect. It should be just (k+3)!
    Thanks from mcolula and topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2013
    From
    Mexico
    Posts
    8
    Thanks
    2

    Re: Mathematical Induction with summations

    Thanks for your answer I just have one doubt more
    doing (k+3)! the common denominator, the resultant expression is

    Mathematical Induction with summations-codecogseqn10.gif

    is there a way in which I can simplify this expression to get this result?

    Mathematical Induction with summations-codecogseqn12.gif
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,792
    Thanks
    1687
    Awards
    1

    Re: Mathematical Induction with summations

    Quote Originally Posted by mcolula View Post
    Thanks for your answer I just have one doubt more
    doing (k+3)! the common denominator, the resultant expression is
    Click image for larger version. 

Name:	CodeCogsEqn10.gif 
Views:	9 
Size:	1.3 KB 
ID:	28566
    is there a way in which I can simplify this expression to get this result?
    Click image for larger version. 

Name:	CodeCogsEqn12.gif 
Views:	9 
Size:	603 Bytes 
ID:	28567
     \begin{align*} \frac{1}{2} - \frac{{n + 1}}{{\left( {n + 2} \right)!}} + \frac{{{{(n + 1)}^2} + (n + 1) + 1}}{{(n + 3)!}} &= \frac{1}{2} + \frac{{ - (n + 1)(n + 3) + {{(n + 1)}^2} + (n + 1) + 1}}{{(n + 3)!}}\\ &= \frac{1}{2} - \frac{{(n + 1) + 1}}{{\left[ {(n + 1) + 2} \right]!}}\end{align*}
    Thanks from topsquark, mcolula and Ragnarok
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2013
    From
    Mexico
    Posts
    8
    Thanks
    2

    Re: Mathematical Induction with summations

    For those who will be visiting this thread in the future this is the whole answer to the problem

    #Problem
    Proof by mathematical induction the following expression

    Mathematical Induction with summations-codecogseqn.gif

    At first place to do this more easy to understand we rewrite this expression like:

    Mathematical Induction with summations-codecogseqn2.gif

    and then we will proof this is true for P(1)

    Mathematical Induction with summations-codecogseqn3.gif

    then we will assume this is true for P(k)

    Mathematical Induction with summations-codecogseqn5.gif

    and then we will proof thi is true for P(k+1)

    Mathematical Induction with summations-codecogseqn6.gif

    then we must obtain that Sk+1 is equal to the sum of Sk and Uk+1

    Mathematical Induction with summations-codecogseqn7.gif

    replacing Sk

    Mathematical Induction with summations-codecogseqn8.gif

    and doing the sum and keeping the common denominator as (k+3)! we obtain this

    Mathematical Induction with summations-codecogseqn13.gif


    then we know that

    Mathematical Induction with summations-codecogseqn15.gif


    the answer continues on the next post
    Attached Thumbnails Attached Thumbnails Mathematical Induction with summations-codecogseqn14.gif  
    Thanks from topsquark and Ragnarok
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    May 2013
    From
    Mexico
    Posts
    8
    Thanks
    2

    Re: Mathematical Induction with summations

    then we obtain this expression

    Mathematical Induction with summations-codecogseqn.gif

    and factoring and doing some algebra

    Mathematical Induction with summations-codecogseqn1.gif

    Mathematical Induction with summations-codecogseqn2.gif

    Mathematical Induction with summations-codecogseqn3.gif

    Mathematical Induction with summations-codecogseqn4.gif

    Mathematical Induction with summations-codecogseqn5.gif

    Now we have proven S is true by mathematical induction.

    I hope this is useful in the future and if there is some kind of language error make know to correct it thanks.
    Last edited by mcolula; June 9th 2013 at 05:45 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 7th 2010, 06:12 PM
  2. Replies: 10
    Last Post: June 29th 2010, 12:10 PM
  3. Mathematical induction
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: November 28th 2009, 01:17 PM
  4. Strong Induction Problem using Summations
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 3rd 2009, 09:12 AM
  5. Mathematical Induction
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: August 12th 2007, 03:45 AM

Search Tags


/mathhelpforum @mathhelpforum