How do you find the recursive formula that describes the sequence 3,7,15,31,63,127.?

Look at the sequence of differences, finding that it is a geometric sequence with common ratio ##2## and hence derive the recursive formula:
##a_1 = 3##
##a_(n+1) = 2a_n + 1##
Write out the original sequence:
##3,7,15,31,63,127##
Write out the sequence of differences of that sequence:
##4,8,16,32,64##
This is a geometric sequence with common ratio ##2##.
Try subtracting it from the original sequence to find:
##-1,-1,-1,-1,-1##
So we can deduce the recursive rule:
##a_1 = 3##
##a_(n+1) = 2(a_n + 1) – 1 = 2a_n+1##
A direct expression for ##a_n## is:
##a_n = 2^(n+1)-1##