What number comes next in the series 3, 7, 15, 31, 63, ?

Explanation

1. 01

   Test the gaps first: 4, 8, 16, 32. They are doubling, which suggests a multiplicative rule.

2. 02

   Check it: each term is double the one before it, plus one. $3 \times 2 + 1 = 7$, $7 \times 2 + 1 = 15$, $15 \times 2 + 1 = 31$, $31 \times 2 + 1 = 63$.

3. 03

   So the next term is $63 \times 2 + 1 = 127$.