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

1 Answer
Oct 30, 2015

Look at the sequence of differences, finding that it is a geometric sequence with common ratio 22 and hence derive the recursive formula:

a_1 = 3a1=3
a_(n+1) = 2a_n + 1an+1=2an+1

Explanation:

Write out the original sequence:

3,7,15,31,63,1273,7,15,31,63,127

Write out the sequence of differences of that sequence:

4,8,16,32,644,8,16,32,64

This is a geometric sequence with common ratio 22.

Try subtracting it from the original sequence to find:

-1,-1,-1,-1,-11,1,1,1,1

So we can deduce the recursive rule:

a_1 = 3a1=3
a_(n+1) = 2(a_n + 1) - 1 = 2a_n+1an+1=2(an+1)1=2an+1

A direct expression for a_nan is:

a_n = 2^(n+1)-1an=2n+11