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##