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

Oct 30, 2015

Look at the sequence of differences, finding that it is a geometric sequence with common ratio $2$ and hence derive the recursive formula:

${a}_{1} = 3$
${a}_{n + 1} = 2 {a}_{n} + 1$

#### 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 $2$.

Try subtracting it from the original sequence to find:

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

So we can deduce the recursive rule:

${a}_{1} = 3$
${a}_{n + 1} = 2 \left({a}_{n} + 1\right) - 1 = 2 {a}_{n} + 1$

A direct expression for ${a}_{n}$ is:

${a}_{n} = {2}^{n + 1} - 1$