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Math Help - Simple arithmetic progression question

  1. #1
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    Simple arithmetic progression question

    I have the answer to this but I'm not sure that I entirely understand it.

    Find the number of terms in the AP: 7 + 9 +...+ (2n+1)


    My solution:

    The ath term will be 2a + 5 so 2a + 5 = 2n + 1 therefore a = n - 2

    This gives me the right answer but what I don't understand is what the difference between n and a is? Initially I thought that they were the same value but of course that is not possible. I would normally think of the 'ath term' as the 'nth term' so is (2n+1) just an expression that happens to use n as a variable just to be awkward?


    I would really appreciate it if somebody could clarify this for me.


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  2. #2
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    Re: Simple arithmetic progression question

    "n" is a specific integer value while you "a" varies from 1 up to n- 2 and so counts the number of terms.

    That is, if n is, say 10, then 7, 9, 11, ..., 2n+ 1 is 7, 9, 11, 13, 15, 17, 19, 21 while a is 1, 2, 3, 5, 5, 7, 8.
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  3. #3
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    Re: Simple arithmetic progression question

    Ok well I normally think of the 'nth' term as the term that is found in a particular place in the sequence/series ie when n=10 the 10th term in the series is 25.

    In this case is n not related to this and just a fixed value? Sorry but I'm still a bit confused...
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  4. #4
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    Re: Simple arithmetic progression question

    Quote Originally Posted by kinhew93 View Post
    Ok well I normally think of the 'nth' term as the term that is found in a particular place in the sequence/series ie when n=10 the 10th term in the series is 25. In this case is n not related to this and just a fixed value? Sorry but I'm still a bit confused...
    I think that the confusion comes from the fact you are using n in two different ways. Let's look at the OP.
    Quote Originally Posted by kinhew93 View Post
    I have the answer to this but I'm not sure that I entirely understand it. Find the number of terms in the AP: 7 + 9 +...+ (2n+1)
    You are to find "the number of terms" as a function of n and that is not the same as the n^{th} term.

    Actually you did the problem correctly. There are n-2 terms in that A.P.

    There are 8=10-2 terms in 7+9+\cdots+(21=2[10]+1).
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  5. #5
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    Re: Simple arithmetic progression question

    Would it be more natural if the problem asked, "Find the number n of terms in the AP: 7 + 9 +...+ (2 a+1) "? Possibly, but this formulation is exactly equivalent to "Find the number a of terms in the AP: 7 + 9 +...+ (2 n+1)." Both variants say, "Fix some number and call it a in the first variant and n in the second variant. Consider the arithmetic progression 7 + 9 +...+ (2 a+1) or, respectively, 7 + 9 +...+ (2 n+1) and find the number of terms as a function of a, or, respectively, n."

    The variable n does not have a monopoly on denoting the index of a sequence. Any variable can be used for this. Similarly, even in a text talking about a sequence of numbers, n can denote many different things. In this case, it denotes some number such that 2n + 1 is the last element of the sequence.
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  6. #6
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    Re: Simple arithmetic progression question

    The AP is: 7,9,11, ( 2n+1)
    Ok Let us presume that the AP has m terms thus we have for the given AP
    First term a1 = 7 common difference d = 2, and last term i.e., mth term = 2n + 1
    But am = a1 + ( m-1) d
    = 7 + ( m-1) x 2 = 7 + 2m 2 = 2m + 5
    It is given that the last term am = 2n + 1
    Thus we have 2m + 5 = 2n + 1
    That give m = n -2
    Hence the AP has n-2 terms.
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