# Thread: Arithmetic & Geometric Progression

1. ## 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!

2. Hello, Tangera!

Here's the first one . . .

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

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

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

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

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