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

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!

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}

• So does the sub-script '+1' actually become a +1 that you add onto the equation? I'm still confused. What's the actual substitution on the LHS though for the second example? @SiongThyeGoh – Curious Aug 13 '17 at 4:52
• The blue color corresponds to the blue color, the red color corresponds to the red color. For the second example, simplify the expression until it is equal to the expression of RHS. – Siong Thye Goh Aug 13 '17 at 5:07
• I know that much but how do I get the LHS to match the RHS. I actually don't understand the substitution. @Siong Thye Goh Is what you wrote the LHS substitution? Because if it is, then it doesn't match the RHS? – Curious Aug 13 '17 at 5:09
• Edited my post. Note that $a^{n+1}=aa^n$. – Siong Thye Goh Aug 13 '17 at 5:11
• Is that for the LHS? @Siong Thye Goh – Curious Aug 13 '17 at 5:27