I am trying to verify a solution I have found to a non-homogeneous difference equation. However, when I substitute this solution into both sides of the difference equation, the two sides are not equal. They are slightly off. Where am I going wrong in the process? Is my solution incorrect? Please see the attached photo below:
Your problem is that you don't understand what a sequence is. It is just a special kind of function, and in your case its domain is $\def\nn{\mathbb{N}}$$\nn$. Your recurrence relation says $x(n+1) = \frac25 x(n) + \frac25$ for every $n \in \nn$. Your solution says $x(n) = -\frac16 (\frac25)^n + \frac23$ for every $n \in \nn$. To check that the solution works, you just need to compute $x(n+1)$ and $\frac25 x(n) + \frac25$ according to the solution for $n \in \nn$ and check that they are equal.
Note that "$x(n) = -\frac16 (\frac25)^n + \frac23$ for every $n \in \nn$" says the same as "$x(k) = -\frac16 (\frac25)^k + \frac23$ for every $k \in \nn$", and means that we can put in any $k$ we like as long as $k \in \nn$, and the equation will hold. For example $x(7) = -\frac16 (\frac25)^7 + \frac23$, because $7 \in \nn$. Now you want $x(n+1)$ where $n \in \nn$, right? $n+1 \in \nn$, so what is $x(n+1)$?