Let $$x_i = \frac{x_{i+1}+x_{i-1}}{2} \quad \text{for $i \in \{2,...,k\}$}$$ and $x_0 = 1$, $x_{k+1}=0$
I want to prove that $x_i = 1-\frac{i}{k+1}$
I noticed that $$x_i = \frac{x_{i+1}+x_{i-1}}{2}=\frac{\frac{x_{i+2}+x_{i}}{2}+\frac{x_{i}+x_{i-2}}{2}}{2}$$ $$\Rightarrow x_i = \frac{x_{i+2}+x_{i-2}}{2}$$
which makes the points equidistant from each other. Visibly, I see this as a linear function if you'd map $(0,x_0),(1,x_2),...$
But I'd still like some tips how to show this in a nice mathematical way.
Certainly noticing that it is a linear function is helpful. You can then make the Ansatz $x_j = Aj + B$ and solve for the coefficients. Alternatively, if you didn't notice that, this is a linear difference equation and the Ansatz $x_j = z^j$ would allow you to find this as well.