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Thread: Number Sequence

  1. #1 Number Sequence 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Recently i have encountered the sequence and series topic, and i am kind of confused with expressing in the nth term.
    can anyone show or explain to me how you could easily express a series of numbers in a simple nth term?

    1)7,5,3,1....... this is an arithmetic sequence so it is easy because you can simply use the formula and you get and geometric sequence is also easy to get.

    But sequences like 4,9,16,25................ how do you get the nth term?
    also some others like 1/2,2/3,3/4,4/5.............
    3,6,7,-2......

    i would appreciate any appropriate suggestions and answers. thank-you


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  3. #2  
    Forum Freshman pbandjay's Avatar
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    Just have to recognize the pattern.

    (4, 9, 16, 25, ...) is a sequence of squares. If n=1 then (n+1)^2 = 2. If n=2 then (n+1)^2 = 9. Etc..., so the nth term in (4, 9, 16, 25, ...) is (n+1)^2.

    (1/2, 2/3, 3/4, 4/5, ...) just has integers in the numerator and denominator that are incremented by 1. The nth term is n/(n+1).

    And I don't yet see a pattern in (3, 6, 7, -2, ...). The online sequence engine doesn't give any straightforward answers either: http://www.research.att.com/~njas/se...lish&go=Search


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  4. #3 Re: Number Sequence 
    . DrRocket's Avatar
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    Quote Originally Posted by Heinsbergrelatz
    Recently i have encountered the sequence and series topic, and i am kind of confused with expressing in the nth term.
    can anyone show or explain to me how you could easily express a series of numbers in a simple nth term?

    1)7,5,3,1....... this is an arithmetic sequence so it is easy because you can simply use the formula and you get and geometric sequence is also easy to get.

    But sequences like 4,9,16,25................ how do you get the nth term?
    also some others like 1/2,2/3,3/4,4/5.............
    3,6,7,-2......

    i would appreciate any appropriate suggestions and answers. thank-you
    One sees that sort of problem frequently, and the expectation is that one will recognize some sort of pattern and from that determine an expression for the "nth term".

    In point of fact there is absolutely no way to know the "nth term" unless it is specified, and the question is poorly formulated. That sort of question is more of a trick than true mathematical question.
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  5. #4  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    strangely enough, these are questions from the last year exam paper.
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  6. #5  
    . DrRocket's Avatar
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    Quote Originally Posted by Heinsbergrelatz
    strangely enough, these are questions from the last year exam paper.
    Whoever wrote the question should have failed.

    This sort of questin is not uncommon. It is not, however, a valid mathematical question.

    Simply writing down the first few terms of a sequence, does not tell you what the next number will be. It could, literally, be anything.

    All of these are valid:

    3,5,7,... ( next nuber is 9 which is the next odd number)

    3,5,7, ... (next number is 11 which the next odd pime)

    3,5,7,... ( next number is 23785 because that is what I wanted to list next and there is no general term for the nth number)
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  7. #6  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Heinsbergrelatz
    strangely enough, these are questions from the last year exam paper.
    Whoever wrote the question should have failed.

    This sort of questin is not uncommon. It is not, however, a valid mathematical question.

    Simply writing down the first few terms of a sequence, does not tell you what the next number will be. It could, literally, be anything.

    All of these are valid:

    3,5,7,... ( next nuber is 9 which is the next odd number)

    3,5,7, ... (next number is 11 which the next odd pime)

    3,5,7,... ( next number is 23785 because that is what I wanted to list next and there is no general term for the nth number)
    Couldn't you say something such as the following:

    The nth term of a series can be determined by the formula ? Namely, providing a formula for the nth term; I suppose this wouldn't be so general as a few (at the very least) would like, however. In the aforementioned example, the first few terms would be this series (starting at one):

    .

    We also sometimes use what's called a common ratio for geometric series to determine the nth term, and in calculus you can even do this for infinite geometric series. A common difference can be used for arithmetic series.
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  8. #7  
    Forum Radioactive Isotope MagiMaster's Avatar
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    According to the Online Encyclopedia of Integer Sequences there are 3364 known answers for 3,5,7,... but it's pretty easy to make up new ones.

    In particular, it's possible to fit an order-3 polynomial to any 3 numbers leaving room to make the next number anything you want.
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  9. #8  
    . DrRocket's Avatar
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    Quote Originally Posted by MagiMaster
    According to the Online Encyclopedia of Integer Sequences there are 3364 known answers for 3,5,7,... but it's pretty easy to make up new ones.

    In particular, it's possible to fit an order-3 polynomial to any 3 numbers leaving room to make the next number anything you want.
    It is trivial to make up new ones. Any number at all can follow 7, and that is the whole point.

    A sequence of integers is a function from the natural numbers to the integers. Any function at all will do.
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