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 2 and hence derive the recursive formula:

a1=3
an+1=2an+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:

a1=3
an+1=2(an+1)1=2an+1

A direct expression for an is:

an=2n+11