How to solve a difference equation in terms of initial value $y_0$ and describe the behavior of its solution as $n$ approaches $\infty$?

Question

Here's the question:

For first equation, the answer should be:

$$y_n = (-1)^n * (.5)^n * (y_0-4) + 4$$ and that $y_n$ goes to $4$ as $n$ approaches $\infty$.

I'm confused on how to get there. If anyone could point me in the right direction to solving these equations that would be much appreciated !!

Answer 1

Try a few terms to see the pattern. Then apply inductively.

$$y_1=-0.5y_0+6 = (-1)^1(0.5)^1(y_0-4)+4$$

$$y_2=-0.5y_1+6 = -0.5(-0.5y_0+6)+6 = (-1)^2(0.5)^2(y_0-4)+4$$

$$\vdots$$

$$y_n = -0.5(y_{n-1})+6 = -0.5[(-1)^{n-1}(0.5)^{n-1}(y_0-4)+4] + 6 = (-1)^{n}(0.5)^{n}(y_0-4) + 4$$

Where the last $+4$ comes from distributing the $-0.5$ to the $+4$ and adding it to $6$.