Suppose that for some positive integer, n, there are n lines in the Euclidean plane such that no two are parallel and no three meet at the same point. Determine the number of regions into which the plane is divided by these n lines.

The Towers of Hanoi game is played with a set of disks of graduated size with holes in their centers and a playing board having three spokes for holding the disks. The object of the game is to transfer all the disks from spoke A to spoke C by moving one disk at a time without placing a larger disk on top of a smaller one. The recursive formula for doing this is for with . Find the closed form for the relationship.

The triangular numbers, allegedly studied by Pythagoras of Samos (c. 572 – 497 BC) and his school of Pythagoreans are given by the number of dots in the following sequence of triangles.

1^{st} triangle 2^{nd} triangle 3^{rd} triangle 4^{th}triangle
How many dots will be in the 5^{th} triangle and 6^{th} triangle?

There are n (n) ovals drawn in a plane. An oval must intersect each of the other ovals at exactly 2 points and no three ovals meet at the same point. Using a recurrence algorithm, develop a conjecture for the number of regions that will be created. (Count the exterior region. Therefore one oval creates 2 regions)

Suppose that for some positive integer, n, there are n lines in the Euclidean plane such that no two are parallel and no three meet at the same point. Predict the number of intersections created by these lines.

Consider the following geometric sequence. Predict the number of black squares in the nth figure.