Geometrical Recursion Activities Kurt and Carol



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Geometrical Recursion Activities

Kurt and Carol
Directions:

  1. Set up a sequence of diagrams depicting dots, squares, regions, lines, intersections, and/or regions.

  2. Find a recurrence formula for this sequence.

  3. Describe this recurrence formula in closed form.

Problems.



  1. 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.




  1. 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.




  1. 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.



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


  1. 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)




  1. 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.




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


…….
1st 2nd 3rd
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