Question

Which of the following is not an example of a recursive sequence?

4, 13, 22, 31, ...
2, 3, 6, 18, ...
6, 9, 15, 24, ...

Answer 1

The sequence 2, 3, 6, 18, ... is not (in the usual simple sense) a recursive arithmetic or geometric sequence.

Reason:
- 4, 13, 22, 31, ... is arithmetic: a1 = 4, a_n = a_{n-1} + 9.
- 6, 9, 15, 24, ... has differences 3, 6, 9, ... so it is recursive: a1 = 6, a_n = a_{n-1} + 3(n−1).
- 2, 3, 6, 18, ... is neither arithmetic (differences 1, 3, 12, ...) nor geometric (ratios 3/2, 2, 3). You also cannot find constants p,q with a_n = p a_{n-1} + q for all terms (solving from the first two terms gives p = 3, q = −3, which fails at the fourth term).

Thus the second sequence is the one that is not an example of a simple recursive sequence of the common types.