What comes next in the sequence 1, 3, 7, 15, 31, 63, _____?

Let us examine the recursive relation between successive terms in the given mathematical sequence:
* $2\times 1+1=3$
* $2\times 3+1=7$
* $2\times 7+1=15$
* $2\times 15+1=31$
* $2\times 31+1=63$

The logic followed is that each term is obtained by doubling the preceding term and adding $1$:

$$T_n=2\times T_{n-1}+1$$

Applying this rule to find the missing next term:

$$\text{Next term}=2\times 63+1=126+1=127$$