If the mean is subtracted from every number in a data set, and the differences are
used to form a new data set, that data set must have a mean:

9514 1404 393

  a[1] = 3

  a[n] = 2×a[n-1] +1

Step-by-step explanation:

We observe the first differences are ...

  7-3 = 4

  15-7 = 8

  31-15 = 16

  63-31 = 32

The first differences increase by a factor of 2, so we expect the recursive formula will have a term that looks like "2×a[n-1]".

The second term is 2·3+1 = 7.

The third term is 2·7+1 = 15.

This suggests our recursive formula is ...

  a[1] = 3

  a[n] = 2×a[n-1] +1