Finding the Rule for a Simple Recurrence Relation

A simple recurrence relation is a relation of the form  
  which use a term to find the succeeding term.
Suppose we are to find the rule for the sequence
3, 7, 15, 31, 63, 127,...
As the sequence progresses each term divided by the previous term approaches 2, so we can write  
  is now easily found by considering the first two terms.
\[7=2 \times 3 +B \rightarrow B=1\]

The rule is  

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