Find a finite summation of a sequence A sequence $x_1 , x_2 , x3 ,..... $ is defined by 
$x_1 = 1$ , $x_{n+1}= (x_n)^2 -k x_n$ , x >1 
Where k is non-zero constant 
$$a)$$
Show that $x_3 = 1-3k + 2k^2$
$$b) $$ Given that $x_3 = 1$ Calculate the value of k , hence find $\sum$ $x_n$ starting from $n=1$ to $n=100$
My turn :
$$x_2 = 1-k , x_3 = (1-k)^2 -k(k-1) = 2k^2 -3k +1 $$
Then 
$$2k^2 -3k = 0 , k = 1.5 $$ But i do not know how can i find the required summation because the sequence is not an arithmetic nor geometric ?
 A: This is a bit of a trick question. Note that since $x_{n+1} = x_n(x_n - k)$, $x_{n+1}$ depends only on $x_n$ and the constant $k$. Thus, if $x_n = x_m$ for some $n \neq m$, then the sequence of values will repeat between those $2$ indices. In particular, you already have $x_1 = x_3 = 1$. With $k = 1.5$, you also get $x_2 = 1 - k = -.5$. Thus, $x_4 = x_3(x_3 - 1.5) = 1(1 - 1.5) = -0.5$, with $x_5 = x_4(x_4 - 1.5) = -0.5(-0.5 - 1.5) = 1$, and so on. 
In general, $x_n = 1$ for all odd $n$ and $x_n = -0.5$ for all even $n$. Thus,
$$\sum_{n=1}^{100} x_n = 50 \times 1 + 50 \times -0.5 = 50 - 25 = 25 \tag{1}\label{eq1}$$
as there are $50$ even values of $n$ and $50$ odd values of $n$ between $1$ and $100$, inclusive.
A: First question: have you computed $x_1$, $x_2$, $x_3$, and $x_4$ when $k = 3/2$?  This would have given some insight.  (Very general strategy: Always be willing to generate the first few values to see if there is an easy to see pattern.)
We already know $k = 3/2$ is special since it makes $x_1 = x_3 = 1$.  Does this force a relation between all the $x_i$ with $i$ odd?  Also, does this force a relation between all the $x_i$ with $i$ even?

    In fact, all the $x_i$ with $i$ even are $-1/2$ and the $x_i$ with $i$ odd are $1$.  This means $$  \sum_{i=1}^{100} x_i = 50 (1 - 1/2) = 25   \text{.}  $$

A: $$x_1 = 1$$
$$x_2 = 1^2-1.5*1=-0.5$$
$$x_3 = (-0.5)^2-1.5(-0.5) = 1$$
$$x_4 = 1^2-1.5*1 = -0.5$$
This is cyclic, with $x_{2k} = -0.5$ and $x_{2k+1} = 1$. Therefore $$\sum_{n=1}^{100} x_n=1*50-0.5*50 = 25 $$
