Consider n circles in a plane. Into how many regions will they divide the plane?
Suppose that n circles divide the plane into rn regions. Now you add another circle, C, that non-trivially intersects each of the first n circles. Start at any point on C and travel around C; you will cross one of the other n circles 2n times.
If p0 and p1 are two consecutive crossings, the arc of C from p0 to p1 divides one of the original rnregions into two smaller regions. You do this 2n times,that gives you a recurrence,
rn+1=rn+some simple function of n,
and you can fairly easily turn it into an expression for rn that is just a summation. If you do it right, it’s a summation that you can evaluate pretty easily in closed form to get a formula for rn.

A circle may be a form consisting of all purposes in a very plane that ar at a given distance from a given point, the centre. Equivalently, it's the curve copied out by a point that moves in a very plane so its distance from a given point is constant.

Main body:

Suppose that n circles divide the plane into rn regions. Now you add another circle, C, that non-trivially intersects each of the first n circles. Start at any point on C and travel around C; you will cross one of the other n circles 2n times.

If p0 and p1 are two consecutive crossings, the arc of C from p0 to p1 divides one of the original rn regions into two smaller regions. You do this 2n times,that gives you a recurrence,

rn+1=rn+some simple function of n,

and you can fairly easily turn it into an expression for rn that is just a summation. If you do it right, it’s a summation that you can evaluate pretty easily in closed form to get a formula for rn.

Hence , the plane divide the region into rn regions.

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