Dear community I would appreciate if you can help me understand these equations. I mean how did he jump from line 1 to line 2? How do u[n] get cancel? Then in the last line where did the "8" come from? And we changed u[n-1] to u[n] once again. Can you please explain it to me? You are my only hope. Thank you.
If I read and understood correctly then the impulse response given in the first line is
$$h[n]=(1-0.9^n)u[n]-(1-0.9^{n-1})u[n-1]\tag{1}$$
Note that for $n=0$ the first expression on the right-hand side of (1) equals $0$ (because $1-0.9^0=0$), so you can rewrite (1) as
$$\begin{align}h[n]&=(1-0.9^n)u[n-1]-(1-0.9^{n-1})u[n-1]\\ &=(1-0.9^n-1+0.9^{n-1})u[n-1]\\ &=0.9^{n-1}(1-0.9)u[n-1]\\ &=0.9^{n-1}\cdot 0.1\cdot u[n-1] \end{align}\tag{2}$$
So if I read the first line correctly, then the second line is wrong. I can't see what is on the left-hand side of the third line and how this relates to the first two lines.