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
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
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 = 2^(n+1)-1