Results 1 to 8 of 8

Math Help - Help required to solve an Arithmatic progression problem

  1. #1
    Newbie
    Joined
    Jun 2011
    Posts
    7

    Help required to solve an Arithmatic progression problem

    The sum of the first p terms of an AP in equal to the sum of the
    first q terms.Prove that sum of the first p+q terms of the same
    AP is zero.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jun 2009
    Posts
    806
    Thanks
    4

    Re: Help required to solve an Arithmatic progression problem

    Sum of p terms S1 = (p/2)[2a + (p-1)d]

    Sum of q terms S2 = (q/2)[2a + (q-1)d]

    Given S1 = S2. Solve the equations and find 2a =

    Then put this value in the sum of p+q terms.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,545
    Thanks
    780

    Re: Help required to solve an Arithmatic progression problem

    Note also that the claim holds if p does not equal q.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jun 2011
    Posts
    7

    Re: Help required to solve an Arithmatic progression problem

    I got 2a+d=0 from the equation S1=S2.But I didn't get how this will help to deduce the sum s(p+q)=0 where s(p+q)=(p+q)/2[2a+(p+q-1)d]

    Please help me
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Help required to solve an Arithmatic progression problem

    Quote Originally Posted by blackhat123 View Post
    The sum of the first p terms of an AP in equal to the sum of the
    first q terms.Prove that sum of the first p+q terms of the same
    AP is zero.
    S_p=S_q

    \frac{(2a_1 +(p-1)d)p}{2}=\frac{(2a_1+(q-1)d)q}{2}

    p(2a_1 +pd-d)=q(2a_1+qd-d)

    2a_1p+p^2d-pd =2a_1q+q^2d-qd

    2a_1p-2a_1q +p^2d -q^2d -pd +qd =0

    2a_1(p-q)+d(p+q)(p-q)-d(p-q)=0

    (p-q)(2a_1+(p+q)d-d)=0

    2a_1+(p+q)d-d=0

    2a_1+((p+q)-1)d=0

    \frac{(p+q)(2a_1+((p+q)-1)d)}{2}=0

    S_{p+q}=0
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6

    Re: Help required to solve an Arithmatic progression problem

    Quote Originally Posted by blackhat123 View Post
    I got 2a+d=0 from the equation S1=S2......
    Then ,you have made some mistake in your calculation. Show us your working so that we can comment on it. Also, remember emakarov's note and use it in the proof.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jun 2011
    Posts
    7

    Re: Help required to solve an Arithmatic progression problem

    Ya... agreed.

    I did some silly mistakes.Now I got it solved.

    Thanks all for your help.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1

    Re: Help required to solve an Arithmatic progression problem

    You could also use

    a+(a+d)+(a+2d)+....+[a+(p-1)d]

    =a+(a+d)+...+[a+(p-1)d]+[a+pd]+[a+(p+1)d]+....+[a+(q-1)d]

    for q>p

    Then

    [a+pd]+[a+(p+1)d]+....+[a+(q-1)d]=0

    Therefore the average must be zero, which is also the average of the first and last terms of this progression.

    0.5[a+pd+a+(q-1)d]=0

    2a+(p+q-1)d=0

    This causes the sum of p+q terms to be zero, since

    S_{p+q}=\frac{2a+(p+q-1)d}{2}(p+q)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Arithmatic Progression Word Problem
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 24th 2009, 10:30 PM
  2. Replies: 1
    Last Post: April 11th 2009, 10:58 PM
  3. Difficult arithmatic progression question
    Posted in the Algebra Forum
    Replies: 4
    Last Post: July 4th 2008, 03:31 AM
  4. arithmatic problem
    Posted in the Algebra Forum
    Replies: 1
    Last Post: June 21st 2008, 12:05 AM
  5. Arithmatic/Geomatrix Progression
    Posted in the Algebra Forum
    Replies: 2
    Last Post: June 2nd 2008, 09:47 AM

Search Tags


/mathhelpforum @mathhelpforum