I found an old book in my uncle's house called Concrete Mathematics, apparently this is a undergraduate-level textbook and I am only in Highschool but I was still curious about the book.
In the 1.3 problem of the book, it wanted to solved recurrent relations without "clairvoyance".
According to the book, the recurrence can be simplified by adding +1 to both sides of the equation. But I don't understand how would +1 simplify the recurrence?
From: \begin{align} T_0 &= 0; \\ Tn &= 2T_{n-1} + 1 \\ \end{align}
To:
\begin{align} T_0 + 1 &= 1; \\ T_n + 1 &= 2T_{n-1} + 2 \\ &= T_{n-1} + 1 + T_{n-1} + 1 \\ \end{align} \begin{align} U_{n} = T_{n-1} \end{align}
Now define $V_n=T_n+1$ and your last equation becomes $$V_n=2V_{n-1}$$ with the solution $$V_n=2^nV_0$$ Adding $1$ makes the constant term disappear.