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 $2$ $2$ and hence derive the recursive formula:

$a_{1} = 3$ $a_1 = 3$
$a_{n + 1} = 2 a_{n} + 1$ $a_(n+1) = 2a_n + 1$

Write out the original sequence:

$3, 7, 15, 31, 63, 127$ $3,7,15,31,63,127$

Write out the sequence of differences of that sequence:

$4, 8, 16, 32, 64$ $4,8,16,32,64$

This is a geometric sequence with common ratio $2$ $2$ .

Try subtracting it from the original sequence to find:

$- 1, - 1, - 1, - 1, - 1$ $-1,-1,-1,-1,-1$

So we can deduce the recursive rule:

$a_{1} = 3$ $a_1 = 3$
$a_{n + 1} = 2 (a_{n} + 1) - 1 = 2 a_{n} + 1$ $a_(n+1) = 2(a_n + 1) - 1 = 2a_n+1$

A direct expression for $a_{n}$ $a_n$ is:

$a_{n} = 2^{n + 1} - 1$ $a_n = 2^(n+1)-1$