Solution of the recurrence relation $y_n = \frac{1}{2} + \frac{1}{2}y_{n-1}$ $y_0 = 0$ and $y_n = \frac{1}{2} + \frac{1}{2}y_{n-1}$. Solution of this reccurent equation is $y_n = 1 - \frac{1}{2^n}$, accordingly with the software. But I do not understand the minus sign since it would be $y_n = \frac{1}{2} + \frac{1}{2}\left(\frac{1}{2}+\frac{1}{2}y_{n-2}\right)$ and so forth until $y_n = \frac{1}{2} + \frac{1}{2^{n}} +x_{0}$.
Can someone try to explain it to me, I'm having a very tough time trying to understand.

$$\begin{split}
y_n &= \frac 1 2 + \frac {y_{n-1}} 2\\
&= \frac 1 2 +\frac 1 2\left( \frac 1 2 +\frac {y_{n-2}} 2\right)\\
&= \frac 1 2 + \frac 1 2 \left(\frac 1 2 +\frac 1 2 \left (\frac 1 2 +\frac {y_{n-2}} 2 \right) \right)\\
&= \dots\\
&= \frac 1 2 + \frac 1 4 + \frac 1 8 +\dots + \frac 1 {2^{n}} + \frac{y_0}{2^n}\\
&= \frac 1 2 \cdot\frac {1-2^{-n}}{1-\frac 1 2} \\
&= 1-2^{-n}
\end{split}$$
