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Math Help - Sum of the first 50 terms

  1. #1
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    Sum of the first 50 terms

    I recently was given a very old Scottish Qualifications Higher paper from 1987. I could do almost every question easily, but when it came to question 10, I was entirely stumped. I have never been taught how to do it (and therefore don't really know what the problem is called so to speak), but here it is anyway. I would be happy if you could shed some light on this.

    10. The sum Sn of the first n terms of an arithmetic series is given by
    S_n = 4n - n^2

    a) Find i) u_1 and u_2, the first 2 terms
    ii) the common difference

    b) Calculate the sum of the first 50 terms of the series
    u_1 + u_3 + u_5 + \ldots
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  2. #2
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    Quote Originally Posted by scotchegg View Post
    I recently was given a very old Scottish Qualifications Higher paper from 1987. I could do almost every question easily, but when it came to question 10, I was entirely stumped. I have never been taught how to do it (and therefore don't really know what the problem is called so to speak), but here it is anyway. I would be happy if you could shed some light on this.

    10. The sum Sn of the first n terms of an arithmetic series is given by
    S_n = 4n - n^2

    a) Find i) u_1 and u_2, the first 2 terms
    ii) the common difference

    b) Calculate the sum of the first 50 terms of the series
    u_1 + u_3 + u_5 + \ldots
    So we know that

    S_1=4(1)-1^2=3=u_1 \implies u_1=3

    and

    S_2=4(2)-2^2=4 =u_1+u_2 \implies u_2=1

    So now we know that the sequence is gives by

    u_n=5-2n

    but we are asked to sum only the first fifty odd terms to get this new sequence we use n=2k-1, k=1,2,3,...

    u_k=5-2(2k-1)=7-4k

    Can you finish from here?
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