can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$
observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer
$t_{0}=0$

The following is a semi-formal variant of induction that is particularly useful for recurrences. 
Let $x_n=2^n-1$. It is easy to verify that $x_0=0$. It is also easy to verify that
$$x_{n+1}=2x_n+1,$$
since $2^{n+1}-1=2(2^n-1)+1$.
So the sequence $(x_n)$ starts in the same way as your sequence and obeys the same recurrence as your sequence. Thus the two sequences must be the same.
