Find $n=N(p)$ for $x_1^2+x_2^2+...+x_n^2 \ge p(x_1x_2+x_2x_3+...+x_{n-1}x_n)$ For what positive integers $n \ge 3$ inequality holds 
$$x_1^2+x_2^2+...+x_n^2 \ge p(x_1x_2+x_2x_3+...+x_{n-1}x_n)$$
for all real numbers, if $p=1; p=4/3; p=6/5$
My work so far: 
a) $p=1 \Rightarrow $for all $n$:
$$x_1^2+(x_1-x_2)^2+(x_2-x_3)^2+...+(x_{n-1}-x_n)^2+x_n^2 \ge 0$$
b) $p=4/3 \Rightarrow n \le 3$
$n=3:$
$$(3x_1-2x_2)^2+x_2^2+(2x_2-3x_3)^2\ge0$$
$n=4$. If $x_1=x_4=2; x_2=x_3=3 -$ inequality is wrong.
c) $p=6/5 \Rightarrow n \le 4$
$n=4:$
$$(3x_1-2x_2)^2+x_2^2+15(x_2-x_3)^2+x_3^2+(3x_3-5x_4)^2\ge0$$
$n=4$. If $x_1=x_5=9; x_2=x_4=15, x_3=16 -$ inequality is wrong.
How to investigate this problem for $p>0$?

The question is equivalent to asking whether the quadratic form given below is positive semidefinite.  The form is  $x^TAx$ for $$A=\left[\begin{matrix}1 & -p/2 & 0 & 0&\cdots & 0&0\\
-p/2 &1&-p/2 & 0& \cdots & 0&0\\
\vdots &&&&\ddots&&\vdots\\
0 & 0 & 0& 0&\cdots& -p/2 &1\end{matrix}\right]$$
where $A$ has $1$ on the main diagonal, and $-p/2$ on the two diagonals immediately above and below.  
This form is positive semidefinite if all of the eigenvalues of $A$ are nonnegative.  Fortunately, the matrix $A$ is tridiagonal and Toeplitz, and therefore has particularly nice eigenvalues:
$$1+2\sqrt{\frac{p^2}{4}}\cos(k\pi/(n+1)),\text{ for }k=1,2,\ldots, n$$
These terms are decreasing as $k$ increases, so we can say that they are all nonnegative if the last one is, that is if $$0\le 1+2\sqrt{\frac{p^2}{4}}\cos(n\pi/(n+1))$$
which we can rearrange as $$0\le 1+|p|\cos\left(\pi \frac{n}{n+1}\right)$$
In particular, if $|p|<1$, then the statement holds for all $n$, as cosine is at most one in absolute value.  If $|p|>1$, then for $n$ sufficiently large the statement will fail to hold, as $\lim_{n\to\infty}\cos\left(\pi \frac{n}{n+1}\right)=-1$, and hence $$\lim_{n\to\infty}1+|p|\cos\left(\pi \frac{n}{n+1}\right)=1-|p|<0$$
