Suppose there are n straight lines on a plane such that every pair of straight lines intersects and no three..?

Suppose there are already n-1 lines, so there are a_(n-1) regions, including of course the infinite regions not enclosed on all sides.
With no line, there is just one region, the whole plane. One line divides the plane into two regions.
i.e. a_0 = 1; a_1 = 2.
When the nth line is drawn, each time it crosses one of the other lines another region is formed, and after tit crosses the (n-1)th line (the final one) there have been n-1 addtional regions formed, plus the extra one formed by this line continuing beyond the (n-1)th line, i.e n extra regions.

Therefore a_n = a_(n-1) + n; or if you prefer, a_(n+1) = a_n + (n+1)
This gives us the sequence 1, 2, 4, 7, 11, 16, ...
which has the differences 1, 2, 3, 4, 5, ...
and the differences between these differences are constant, all being 1, therefore
a_n = (1/2)n² + pn + q
Since a_0 = 1, we get q = 1.
And a_1 = 2, i.e. (1 /2) + p + 1 = 2
Therefore p = (1/2)
and so
a_n = (1/2)n² + (1/2)n + 1

this may be a real assertion. listed under are the three disjoint (at the same time unique) gadgets: a million. The factors on the line 2. The factors on one area of the line 3. The factors on the different area of the line. factors have not have been given any length, in basic terms place, for this reason each and each disjoint set has a limiteless sort of people.