Square Root of Two

From Prime Obsession by John Derbyshire. Consider the sequence 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ....

Each term in the sequence gives a better approximation to the square root of two. The rule is add top and bottom to get new bottom, add top and twice bottom to get new top.

Homework assignment: prove the sequence converges to the square root of two.

Another cool way to get an approximate value for the square root of two is to enter 2 in your calculator and hit the square root key.

P. S. If Hoagie gives a correct proof, keep in mind he has a Ph.D. in statistics.