# Induction Examples

(from http://www.math.sc.edu/~sumner/numbertheory/induction/Induction.html)

1. Prove that for every $n\ge{1}$, 133 divides

$\huge{11^{n+1}+12^{2n-1}}$

2. An integer n is a perfect square if it is the square of some other integer.

(For example 1, 4, 9, 16, 25 and 36 are all perfect squares.) Prove by induction that the sum 1 + 3 + 5 + 7 + ... + 2n-1 (i.e. the sum of the first n odd integers) is always a perfect square.

Can you give a "geometric proof"?

3. Show that for $n\ge{1}$ any 2n x 2n board with one square deleted can be covered by Triominoes.