1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Arithmetic progression sum and nth term

  1. Feb 17, 2015 #1

    Suraj M

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    The ratio of sums of 2 AP for n terms each is ## \frac{3n + 8}{7n + 15}##
    that is
    $$ {\frac{s_a}{s_b}} = \frac{3n + 8}{7n + 15} $$
    find the ratio of their 12th terms.
    $$ Required= \frac{a₁_a+(n-1)d_a}{a_b + (n-1)d_b}$$


    2. Relevant equations
    Tn = a + (n-1)d


    3. The attempt at a solution
    OKay the regular way that gives the right answer is this..
    $$ {\frac{2a_a + (n-1)d_a}{2a_b + (n-1)d_b} } = {\frac{3n + 8}{7n + 15}} ~~ eq1 ~~$$
    $$required = \frac{a_a+11d_a}{a_b+11d_b} $$
    so putting n = 23 in the eq 1 we get the answer as ## \frac{7}{16} ##
    but i tried another method..
    $${\frac{s_a}{s_b}} ={ \frac{3n + 8}{7n + 15}}$$
    here we can say..
    $$ s_a = (3n + 8)x $$
    $$ s_b = (7n + 15)x $$
    if i put n = 1 ,2 ,3 for both these AP's
    for AP 1: using sa
    s₁=a₁ right??
    then a₁=11x
    then s₂=14x so a₂ = s2 -s1 = 3x
    then s3 =17x so a3 = s3-s2 = 3x
    how is a2=a3 ??
    similarly even for AP2
    but why??
     
    Last edited: Feb 17, 2015
  2. jcsd
  3. Feb 17, 2015 #2

    Svein

    User Avatar
    Science Advisor

    deleted...
     
    Last edited: Feb 17, 2015
  4. Feb 17, 2015 #3

    Svein

    User Avatar
    Science Advisor

    OK. Let AP1 be given as [itex] a_{n}[/itex] and AP2 be given as [itex] b_{n}[/itex] Then the sums are given as [itex]s_{a,n}=\frac{n(a_{1}+a_{n})}{2} [/itex] and [itex]s_{b,n}=\frac{n(b_{1}+b_{n})}{2} [/itex]. The ratio is therefore [itex]\frac{s_{b,n}}{s_{a,n}}=\frac{(b_{1}+b_{n})}{(a_{1}+a_{n})} [/itex]. Substitute the expression for [itex] a_{n}[/itex] and [itex] b_{n}[/itex]: [itex]\frac{s_{b,n}}{s_{a,n}}=\frac{(b_{1}+b_{1}(n-1)e)}{(a_{1}+a_{1}(n-1)d)}=\frac{3n+8}{7n+16}[/itex]. Solve for [itex] \frac{b_{1}}{a_{1}}[/itex] and insert...
     
  5. Feb 17, 2015 #4

    Suraj M

    User Avatar
    Gold Member

    No Svein, i know how to get the answer, my question is why am i getting the terms after the first term, equal by the method I've shown above?
     
  6. Feb 17, 2015 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I think you are confusing yourself with your notation. You say
    Initially, you wrote that s1 and s2 were two different arithmetic progressions. Now you are using "s1" and "s2" to mean the first two partial sum of one arithmetic progression.
     
  7. Feb 17, 2015 #6

    Suraj M

    User Avatar
    Gold Member

    Ok fine, just notation mistake. check the original post, now ok?
    let those s1 s2 s3 be fore the first AP only.
    then a2=a3=... why?
     
  8. Feb 24, 2015 #7

    Suraj M

    User Avatar
    Gold Member

    Hello? anyone?
     
  9. Feb 25, 2015 #8

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You are assuming x is a constant, independent of n. It need not be. It is only necessary that it is the same function of n in nu numerator and denominator.
     
  10. Feb 25, 2015 #9

    Suraj M

    User Avatar
    Gold Member

    x is the common factor, it must be constant right?
     
  11. Feb 25, 2015 #10

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Why might it not be, say, n? I.e., sa = (3n+8)n, etc.
     
  12. Feb 26, 2015 #11

    Suraj M

    User Avatar
    Gold Member

    ohh, did not not think of that, sorry, but then the common factor may be any function of n. Is there no way of finding that common factor between sa and sb?
     
  13. Feb 26, 2015 #12

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You know that sa is the sum of an AP, so it must be a quadratic function of n. You also know one of its factors.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Arithmetic progression sum and nth term
  1. Arithmetic progression (Replies: 2)

  2. Arithmetic Progression (Replies: 1)

  3. Arithmetic progression (Replies: 9)

  4. Arithmetic progression (Replies: 26)

Loading...