How to verify a solution of a difference equation by substitution?

Question

so I'm finding trouble in verifying a solution I have found for first order homogeneous and inhomogeneous difference equations. I can find the solutions fine, but it is verifying them through substitution (i.e LHS = RHS) that I have trouble with. Especially with what to do on the LHS.

1. Where does the $+1$ come from on the left-hand side? I feel like it is coming from nowhere?
2. For the second example, what do I do on the left-hand side? Substitute my solution into the left-hand side? I just put it there and then what? How does it change? I am so confused.

Thanks so much to anyone who can help me!

Answer 1

For both example, you have to know what is the solution that $x_n$ represents.

For the first example, $x_{\color{blue}n} = 2\color{blue}{n}+\color{red}{1}$

Hence $$x_{\color{blue}{n+1}}=2(\color{blue}{n+1})+\color{red}{1}$$

For the second example. $x_{\color{blue}n} = \frac14 \left( \frac15\right)^{\color{blue}n}+\frac14$

Hence $$x_{\color{blue}{n+1}} = \frac14 \left( \frac15\right)^{\color{blue}{n+1}}+\frac14$$

Edit:

\begin{align}x_{\color{blue}{n+1}} &= \frac14 \left( \frac15\right)^{\color{blue}{n+1}}+\frac14 \\ &= \frac14 \frac15\left( \frac15\right)^{n}+\frac14 \\ &=\frac1{20}\left( \frac15\right)^{n}+\frac14\end{align}