I found this question on a GRE practice exam. It is stated as follows.
Let $f$ be a real-valued function defined on the set of integers and satisfying $f(x)= \frac{1}{2}f(x-1)+\frac{1}{2}f(x+1)$. Which of the following must be true?
I. The graph of $f$ is a subset of a line.
II. $f$ is strictly increasing.
III. $f$ is a constant function.
My approach was to plot points on a graph. First I choose 2 arbitrary points for $f(x-1)$ and $f(x+1)$. Then, $f(x)$ is the midpoint between $f(x-1)$ and $f(x+1)$ while $x$ is the midpoint between $x-1$ and $x+1$. These three points lie on a line which led me to believe that I. is true. If I choose $f(x-1)$ to be greater than $f(x+1)$, then the three points will be a subset of a line with negative slope, which would rule out II. and III.. Since what I've done isn't very rigorous, I am wondering if my reasoning is correct or if there is an easier/faster way to look at this problem. Thanks in advance.
Here's how I look at it: after some rearrangement you have that $$f(x+1) - f(x) = f(x) - f(x-1)$$ Or $\Delta {f}_{x+1} = \Delta f_x$ where $\Delta f_i$ is the change in $f$ between $i-1$ and $i$. Since the function is increasing by the same amount every step, it must be a subset of a line.