Results 1 to 3 of 3

Math Help - [SOLVED] induction proof of summation

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    80

    [SOLVED] induction proof of summation

    use mathematical induction to show that the summation from i=n to 2n-1 (2i+1) = 3n^2 whenever n is a nonnegative integer
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Quote Originally Posted by jmedsy View Post
    use mathematical induction to show that the summation from i=n to 2n-1 (2i+1) = 3n^2 whenever n is a nonnegative integer
    Let P(n) be the statement that P(n)=\sum_{i=n}^{2n-1}2i+1=3n^2, for all non-negative n. Assume that P(n) is true for some n \le k

    P(k+1)=\sum_{i=k+1}^{2(k+1)-1}2i+1

    The upper limit of the sum is also 2k+1. Think about how this second sum compares to the first.

    \sum_{i=k+1}^{2(k+1)-1}2i+1=\left(\sum_{i=k}^{2k-1}2i+1\right)-(2k+1)+(4k+1)+(4k+2+1)

    I'll let you figure out why P(k+1) can be written like that. What extra terms does it contain that P(k) does not?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2009
    Posts
    80
    Quote Originally Posted by Jameson View Post
    Let P(n) be the statement that P(n)=\sum_{i=n}^{2n-1}2i+1=3n^2, for all non-negative n. Assume that P(n) is true for some n \le k

    P(k+1)=\sum_{i=k+1}^{2(k+1)-1}2i+1

    The upper limit of the sum is also 2k+1. Think about how this second sum compares to the first.

    \sum_{i=k+1}^{2(k+1)-1}2i+1=\left(\sum_{i=k}^{2k-1}2i+1\right)-(2k+1)+(4k+1)+(4k+2+1)

    I'll let you figure out why P(k+1) can be written like that. What extra terms does it contain that P(k) does not?
    Thanks. Instead of using the appropriate arbitrary k, I've been using n+1 (which proves the proposition for the value sequentially after n, but not for all n). Anyway, the steps of this particular proof are clear now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Need some help with an induction proof
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 25th 2010, 08:28 AM
  2. Integral/summation proof via induction
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 5th 2010, 10:09 AM
  3. Induction proof of summation
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 21st 2010, 10:56 AM
  4. Proof of summation of i^3 by induction, help needed
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 20th 2009, 12:24 AM
  5. [SOLVED] induction proof
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 9th 2008, 07:44 PM

Search Tags


/mathhelpforum @mathhelpforum