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

1 Answer
Oct 30, 2015

Answer:

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#

Explanation:

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#