Results 1 to 6 of 6

Math Help - Easy problems

  1. #1
    Member
    Joined
    Dec 2005
    Posts
    117

    Easy problems

    Show that the sum of the first n positive odd integers is n^2.

    Show that the sum of the first n positive even integers is n^2+n
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    You have,
    1+3+5+...+(2n-1)
    Or,
    (2\cdot 0-1)+(2\cdot 1-1)+...+(2\cdot n -1)
    Regroup,
    2(0+1+2+...+n)-(1+1+...+1)
    Using sum formulas,
    2\cdot \frac{n(n+1)}{2}-n
    Thus,
    n^2+n-n=n^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2005
    Posts
    117
    Quote Originally Posted by ThePerfectHacker
    You have,
    1+3+5+...+(2n-1)
    Or,
    (2\cdot 0-1)+(2\cdot 1-1)+...+(2\cdot n -1)
    oops, i meant to post this is the calculus section, but i guess it can apply to number theory.

    why (2\cdot 0-1)? This evaluates to -1, and we're looking for the positive integers.
    Last edited by c_323_h; May 1st 2006 at 08:03 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Apr 2006
    Posts
    401
    (2n-1) for definition of odd; start the summation from 1...infinity.

    Or, alternatively, 2n+1, and starting it from 0...infinity.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    You can easily prove these two identities with
    Mathematical Induction

    I was just too lazy to use induction on these two problems. And decided to use a more elegant way.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    627
    Hello, c_323_h!

    Show that the sum of the first n positive odd integers is n^2.
    We have: . 1 + 3 + 5 + \hdots + (2n-1)

    . . an Arithmetic Progression with first term a = 1 and common difference d = 2.

    The sum of the first n term is an A.P. is: . S_n\;=\;\frac{n}{2}[2a + d(n-1)]

    So we have: . S\;=\;\frac{n}{2}[2\cdot1 + 2(n-1)]\;=\;n^2


    Show that the sum of the first n positive even integers is n^2+n
    We have another A.P. . 2 + 4 + 6 + \hdots + 2n

    . . with first term a = 2 and common difference d = 2.

    The sum is: . S\;=\;\frac{n}{2}[2\cdot2 + 2(n-1)] \;= \;n(n + 1)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Need help with a few easy calculus problems
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: April 24th 2010, 02:14 PM
  2. Can someone help with these problems. They're easy
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: August 18th 2009, 07:26 AM
  3. Help on Easy Slope Problems
    Posted in the Algebra Forum
    Replies: 4
    Last Post: May 13th 2009, 08:32 PM
  4. just a few easy problems...
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 24th 2008, 03:16 PM
  5. Probably 2 very easy probability problems
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: May 15th 2008, 02:09 AM

Search Tags


/mathhelpforum @mathhelpforum