Hey there! Sign in to join this conversationNew here? Join for free

geometric series and arithmetic series Watch

Announcements
    • Thread Starter
    Offline

    13
    ReputationRep:
    hi guys!

    i've tried question 23 multiple times with attempts to work out r, but i cannot get a numerical value for it as all my equations for r = have two other unknowns (a and d) within them.

    how can a question like this be solved? thank you Name:  image.png
Views: 9
Size:  65.6 KB
    • Very Important Poster
    Online

    21
    ReputationRep:
    (Original post by ashaxo99)
    hi guys!

    i've tried question 23 multiple times with attempts to work out r, but i cannot get a numerical value for it as all my equations for r = have two other unknowns (a and d) within them.

    how can a question like this be solved? thank you Name:  image.png
Views: 9
Size:  65.6 KB
    Please post your working - perhaps your most recent attempt
    Offline

    10
    ReputationRep:
    I have devised a solution. Can you tell me what the correct answer is and if I'm right, I'll explain it.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by Kevin De Bruyne)
    Please post your working - perhaps your most recent attempt
    my apologies, i should've included this in my main post my working is really messy and probably won't make a lot of sense at all, but i've basically been trying to solve for r, but not with much luck! Name:  image.jpg
Views: 8
Size:  357.8 KB
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by thekidwhogames)
    I have devised a solution. Can you tell me what the correct answer is and if I'm right, I'll explain it.
    thank you for taking the time to do so! the correct answer is D) 1/4
    Offline

    14
    ReputationRep:
    (Original post by ashaxo99)
    thank you for taking the time to do so! the correct answer is D) 1/4
    ar = a+4d (1)
    ar^2 = a+5d (2)

    Subtract (1) from (2) to get d in terms of a and r.

    Substitute for d in (1).
    Offline

    10
    ReputationRep:
    (Original post by ashaxo99)
    thank you for taking the time to do so! the correct answer is D) 1/4
    This was quite a nice problem - is it for A level (if so it's quite challenging).

    Anyhow, my answer was correct. Here's how I did it:

    Spoiler:
    Show

    Given firstly all the terms are different (note this).

    We know:

    - both first terms are the same
    - ar = a+4d
    - ar^2 = a+5d (I)

    ar=a+4d
    ar^2=a+5d

    Add d to the first equation yields ar+d=a+5d and that's equal to the same thus the expressions must obviously be equal. That is, ar+d=ar^2.

    Then by simple algebra and factorising we get d=ar(r-1)

    Let's put this back into (I); ar^2 = a+5ar(r-a)

    So we know ar^2 = a+5ar^2 - 5ar

    Putting it all to one side gives us 4ar^2-5ar+a=0

    Factoring gives us (4ar-a)(r-1) = 0

    So r can be 1 however that makes all the terms the same hence this solution for is rejected.

    4ar-a=0 --> 4ar=a --> 4r=1 --> r=1/4

    Hence r=1/4 and thus answer is (D).

    Lovely question.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by tiny hobbit)
    ar = a+4d (1)
    ar^2 = a+5d (2)

    Subtract (1) from (2) to get d in terms of a and r.

    Substitute for d in (1).
    thank you very much!
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by thekidwhogames)
    This was quite a nice problem - is it for A level (if so it's quite challenging).

    Anyhow, my answer was correct. Here's how I did it:

    Spoiler:
    Show


    Given firstly all the terms are different (note this).

    We know:

    - both first terms are the same
    - ar = a+4d
    - ar^2 = a+5d (I)

    ar=a+4d
    ar^2=a+5d

    Add d to the first equation yields ar+d=a+5d and that's equal to the same thus the expressions must obviously be equal. That is, ar+d=ar^2.

    Then by simple algebra and factorising we get d=ar(r-1)

    Let's put this back into (I); ar^2 = a+5ar(r-a)

    So we know ar^2 = a+5ar^2 - 5ar

    Putting it all to one side gives us 4ar^2-5ar+a=0

    Factoring gives us (4ar-a)(r-1) = 0

    So r can be 1 however that makes all the terms the same hence this solution for is rejected.

    4ar-a=0 --> 4ar=a --> 4r=1 --> r=1/4

    Hence r=1/4 and thus answer is (D).

    Lovely question.

    ah, what a stunning question indeed

    thank you so much for your extremely comprehensive reply! that was super helpful the question is from a cambridge entrance exam for economics! last year's, actually! once again, thank you for taking the time to devise and explain your solution
    Offline

    12
    ReputationRep:
    What is this? Is it the TMUA or something else?
    Posted on the TSR App. Download from Apple or Google Play
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by Kaiylar)
    What is this? Is it the TMUA or something else?
    it's from the economics admissions assessment for Cambridge, although the maths questions on it are extremely similar to the TMUA
    Offline

    12
    ReputationRep:
    (Original post by ashaxo99)
    it's from the economics admissions assessment for Cambridge, although the maths questions on it are extremely similar to the TMUA
    Ah right.
    Posted on the TSR App. Download from Apple or Google Play
    Offline

    10
    ReputationRep:
    (Original post by ashaxo99)
    ah, what a stunning question indeed

    thank you so much for your extremely comprehensive reply! that was super helpful the question is from a cambridge entrance exam for economics! last year's, actually! once again, thank you for taking the time to devise and explain your solution
    No problem, it was good practice for my Maths skills too. Good luck!
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What time of year is the worst for students?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.