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
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
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=2n+1−1