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Math Help - Arithmetic & Geometric Progression

  1. #1
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    Arithmetic & Geometric Progression

    Thank you for helping!!

    Q1: An arithmetic progression has n terms and common difference d, where d > 0. Prove that the difference of the sum of the last k terms and the sum of the first k terms is (n - k)kd.

    Q2: Given that p = (x^2) - 2x - 1, q = (x^2) + 1, r = (x^2) + 2x -1, find all the real values of x for which
    i) p, q, r are in arithmetic progression;
    ii) (p^2), (q^2) and (r^2) are in geometric progression.

    Q3: Given that Sn denotes the sum of the first n terms of a certain arithmetic progression in which the common difference is not zero and that S2n = kSn for all values of n, find the value of the constant k.

    Thank you very very much!
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  2. #2
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    Hello, Tangera!

    Here's the first one . . .


    Q1) An A.P. has n terms and common difference d, where d > 0. .Prove that the difference
    of the sum of the last k terms and the sum of the first k terms is: (n - k)kd
    The sum of the first k terms is: . S_k \:=\:\frac{k}{2}[2a + (k-1)d] \;=\;ak + \frac{dk^2}{2} - \frac{dk}{2}\;\;{\color{blue}[1]}


    The last k terms are: . \begin{Bmatrix}a + (n-k)d \\ \vdots \\ a + (n-3)d \\ a + (n-2)d \\ a + (n-1)d \end{Bmatrix}

    Their sum is: . S_L\;=\;ka + \left(kn - \frac{k(k+1)}{2}\right)d \;=\;ak + dkn - \frac{dk^2}{2} - \frac{dk}{2}\;\;{\color{blue}[2]}


    Subtract [2] - [1]: . \left[ak + dkn - \frac{dk^2}{2} - \frac{dk}{2}\right] - \left[ak + \frac{dk^2}{2} - \frac{dk}{2}\right]

    . . . . . . . . . . . = \;\;dkn - dk^2 \;\;=\;\;(n-k)dk

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