Find the next number in the given series: $3,7,15,31,63,\dots$

Let's look at the difference between consecutive terms in the series:
* $7-3=4=2^2$
* $15-7=8=2^3$
* $31-15=16=2^4$
* $63-31=32=2^5$

The differences are powers of $2$, doubling each time ($4,8,16,32$). Following this pattern, the next difference should be $32\times 2=64$.

$$\text{Next term}=63+64=127$$

Alternative logic: Every subsequent term is calculated as $(\text{current term}\times 2)+1$:

* $(3\times 2)+1=7$
* $(7\times 2)+1=15$
* $(15\times 2)+1=31$
* $(3\times 2)+1=63$
* $(63\times 2)+1=127$

Both logic branches yield $127$.