Moser's Circle Problem

Take two points on a circle,
and draw a line straight through.
The space that was encircled
is divided into two.

To these points add a third one,
which gives us two more chords.
The space through which these lines run
has been fissured into four.

Continue with a fourth point,
and three more lines drawn straight.
The new number of disjoint
regions sums, in all, to eight.

A fifth point and its four lines
support this pattern gleaned.
Counting sections, one divines
that there are now sixteen.

This pattern here of doubling
does seems a sturdy one.
But one more step is troubling
as the sixth gives thirty one.