I'm watching this series but the problem is that I don't know how step 1 gets to step 2 and from step 2 to step 3.
Could someone tell me what is done to get to those steps? I do not get it.
Step 1: $$t(n-2)=\frac{n-1}{n} *\left(\frac{n-2}{n-1} *\left(\frac{n-3}{n-2} * t(n-2-1)+1\right)+1\right)+1 $$ Step 2: $$t(n-2)=\frac{n-3}{n} * t(n-3)+\frac{n-2}{n-1} * \frac{n-1}{n}+\frac{n-1}{n}+1$$ Step 3: $$t(n-2)=\frac{n-3}{n} * t(n-3)+\frac{n-2}{n} * \frac{n-1}{n}+1$$
Thank you!
$$\left(\frac{n-1}{n}\right)\left(\frac{n-2}{n-1}\left(\frac{n-3}{n-2}\times t(n-3)+1\right)+1\right)+1\\ \implies \frac{n-1}{n}\left(\frac{n-3}{n-1} \times t(n-3)+\frac{n-2}{n-1}+1\right)+1 \\ \implies \frac{n-1}{n}\left(\frac{n-3}{n-1} \times t(n-3)+\frac{n-2}{n-1}+1\right)+1\\ \implies \frac{n-3}{n}\times t(n-3)+\frac{n-2}{n-1} \times \frac{n-1}{n}+\frac{n-1}{n}+1$$
Sorry for repeating the step
And in your question the third step written is wrong. You multiplied $\frac{n-2}n$ and $\frac{n-1}n$, you have to add it.