Quadratic Sequences



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38 Responses

  1. john says:

    when i first saw this i thought that i was never going to understand it since im hopeless at maths.
    but really , this site is great and i really do understand quadratic equations now !!

  2. mking says:

    but what if there is more than one nth term

  3. Hey,
    Thanks for the article! Very helpful!
    I was wondering what level this work is? And is there another quadratic sequence where you have to find a b and c?
    Please help.
    Lucie Mcroosy

    • Author says:

      Hello Lucie,

      Glad you liked the article.

      This level is what 18-19 would probably do. It is college level maths or equivalent to A level.

      I didn’t quite understand your last question.

      • Jason says:

        This is for 18-19 years old students
        wow,-i’ve fairly recently started year 8 and we’re learning this- i came here for help with my homework i had to find the nth term of two quadratic sequences
        1))0,5,12,21,32-Which i think the answer is n2+2n-3
        I’m having a lot of trouble with Q.2
        PLEASE HELP!

        • Jason says:

          just ignore my last comment :( ive read on

          • Author says:

            Hello Jason,

            Yes the first one is correct:
            [ n^2 + 2n -3 ]
            Are you sure you have written the second sequence correctly?

            May be the sequence is:
            [ 0, 3, 10, 21, 36 ]
            So then you will get;
            [ 2n^2 – 3n + 1 ]
            I thought I add the answer for you as well since you have spent a lot of time on it.

  4. Hello Author,

    Thanks for your answer. Don’t worry about my last question, I was probably dreaming.
    Thanks ever so much,
    Lucie Mcroosy.

  5. Jon says:

    This is only level 7/8 maths at best. That is equivalent to C/B grade at GCSE. However, you would need to have something that includes a more complicated equation so that you could find “b” as well as “c” for the formula. I’ve just taught this to my year 8’s (12 year olds) but they are top set.

  6. Author says:

    Thanks for the reply Jon,

    Your answer is better. I wasn’t sure this was in GCSE and 12 years old exams.

  7. Hey there Mr Jon
    Well, I am 12 and second to top set. Top set does level 8, second does level 7. There are 6 sets altogether. We’ve learnt about the a b c. I think ‘a’ its a half the answer from the differences of the differences. ‘C’ is find the 0 term number. And I think ‘b’ is the hardest one-i think you have to do some simplifying and replacing the N’s with term number 1?
    Am I wrong? I’ve kinda forgotten.
    Lucie Mcroocy :-) :-) :-)

  8. Anonymous says:

    Hi, What is the second difference is a constant but it is -5. for example,

    Terms 4, -1, -11, -26, -46
    Difference -5, -10, -15, -20
    2nd Diff -5, -5, -5

    How would N to the power of 2 come into this? Help. thanks.

  9. Kaitlin says:

    I really don’t understand why the n^ thing changes depending on the number? i’ve got online maths homework to do and i have to get over 80% and im just so confused:(((( This helped me a bit but im still so confused

    • megan says:

      YOu posted that 2 years ago but for each term the number n changes so if you want the 100th term, n is 100

  10. mxxxxxxxxx says:

    link to a good online calculator for this??

  11. chasm says:

    the question is
    12 , 25 , 42 , 63 , 88
    can u pls help me find the sequence ?

    • Author says:

      Hello Chasm,

      You must show that you have tried so that we know where you’re stuck. Have you tried it? So you will need to find the first difference. You do that by subtracting the previous term from the next term. For example:
      [ 25-12=13, 42-25=17, 63-42=21, 88-63=25 ]
      So now you know that the first difference is:
      [ 13, 17, 21, 25 ]
      Using the examples above, I am sure you can continue from here.

  12. *** says:

    Hi, what about the sequence : 3,10,21,36,55 ?

  13. Reliable says:


    The example that you’ve shown is to find the nth term of nonlinear sequence with positive second difference. How about sequence with negative second difference? Like this………..

    Terms: 36, 35, 32, 27, 20, 11
    Difference: -1, -3, -5, -7, -9
    2nd difference: -5, -5, -5, -5

    How to find?

  14. Helena says:

    I have this sequence: 2.7,15,26..
    The second difference is 3 and I’m not sure what to do?!

  15. Veeti Kuivalainen says:

    I have worked on this equation for the past hour.
    I am in Grade 9 High Level IGCSE Maths, and I can’t do this equation. I know the second difference is -4, and so according to Un=an^2+bn+c, a should be -2. But I can’t figure out b or c; a+b is supposed to be 4 (the first difference) and a+b+c should be 9 (the first term) and so b SHOULD be 6 and c SHOULD be 5 but it simply doesn’t seem to match up to the equation. I really have tried this all, but it isn’t working. PLEASE HELP! Our teacher likes to challenge us, and I’ve done the rest of the homework but this one has got me completely stuck.

  16. ELMA JOPIA says:

    There is another way to solve this quadratic sequence faster and easier.

  17. hannah says:

    hi whats the nth term of -1, 3, 11, 23, 39? i have tried but cant work it out?

  18. is 0,1,4,9 a quadratic sequence

  19. elliot says:

    Good stuff

  20. martin kennedy says:

    There are two mistakes in the examples – the first difference in 1,1,3,7,13 – is 0,2,4,6 – and 5,21,47,83,128 – should read 5,21,47,83,129

  21. Lucy says:

    hi. I have been looking for the answer several times but I couldn’t figure it out.
    What is the nth term for 0,1,6,15 28.

  22. Ellie says:

    I have to find the nth term (the formula) for 84,60,40,24,12,4 and i can see that the second difference is 40 so it would be 2nsquared and then following the rest of your method i get 2nsquared – 30 but this doesn’t work? is it because it is decreasing not increasing can someone help please?

  23. Malloney says:

    I was wondering as to how to find the sequence rule in terms of n, when the second difference is a multiplied constant pattern (rather than an added one)

    for example
    1, 4, 16, 64, … (first term=4, second term = 4, etc)
    [First difference: +3, +12, +48. Therefore, the second difference can be: x4, x4, x4]

    it should be something along the lines of n^4, i figured at first… but i can’t seem to figure it out.

    Help would be appreciated, thanks.

  24. Rika says:

    I am a Mathematics teacher and this is a new meyhod! Amazing that there is always something to learn. But is there not a msistake with example: 1; 1; 3; 7; 13: The first difference iss 0; 2; 4; 6?

  25. Dana says:

    Alright, so I was given 2, 3, 7, 16, 32, 57….
    I had to figure out the next two terms which was easy and then put it into the quadratic formula, which was not so easy…
    So far my formula is:
    0.5x^2 + ?n +5

    I really am at a loss to figure out b and I need this assignment is as soon as possible and I was wondering if you would be able to help thank you so much !

  26. Hi Dana – This is a cubic sequence in the form of an^3+bn^2+cn+d – plug in the first four terms and solve as multiple linear equations by elimination for the four unknowns and you get n^3/3-n^2/2+n/6+2
    Check the link for a detailed explanation/example

  27. Brian says:

    Hi I have found your artical very helpful but I have one confusion it is with the example – find the nth term for this sequence 1, 1, 3, 7, 13 I can’t see how you you worked out the first difference is this a typo or am I missing something because the difference between 1, and 1 aint 3 likewise 1, and 3 aint 5 nor 3,and 7 7 help me clear up this confusion. Cheers.

  28. Ginaro says:

    I was asked to find the n th term for the sequence 4, 12, 24, 40. I figured out that the second difference is 4 but I cant seem to figure out the n th term! Could you help me out please?
    Thank you

  29. zoe says:

    really stupid question but what happens if you have a second difference of a number such as 3? As this doesnt easily divide into 2? Does this still count as a quadratci equation?

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