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

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Answer 1 · 2015-10-30T08:42:22

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:

$a_n = 2^(n+1)-1$