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QUESTION

# 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