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

1 + 3 + 5 + ... + (2n - 1) =
n^{2}

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(N^{2}).
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 2^{2n+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 2^{n+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.]