<s>[INST] 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.
 [/INST] Repeat the formula $n$ times: $y_n  = \dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots + \dfrac{1}{2^{n-1}} + \dfrac{1}{2^n}+\dfrac{1}{2^n}y_0= 1 -\dfrac{1}{2^n}$
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