Proof by Induction practice problems

1. Use proof by induction to prove that, for all n ≥ 1

1 + 3 + 5 + ... + (2n - 1) = n2

2. An H Tree is a way of drawing a perfect binary tree such that the area covered by the tree with N nodes is O(N2).  Generation 0 of an H-tree is a single node, the root.  Each subsequent generation grows the tree by two levels, arranging the sub-trees in the form of the letter H.

Here is a diagram showing the H-trees for generations 0, 1, 2, and 3:


Assume that the nodes of an H-tree are points (0 area).  Also assume that all "wires" (straight lines) are at least 1 unit of distance away from any node that the wire doesn't touch.

(a) Use proof by induction to prove that a generation n H-tree has 22n+1- 1 nodes for all n ≥ 0.

(b) [Harder] Use proof by induction to show that the the length of one side of the square which bounds a generation n H-tree is 2n+1 - 2.  [Hint: a generation n+1 H-tree is built out of 4 copies of the generation n H-tree.  There is a gap of 2 units of distance between each copy.]