sqrt2

mathematical nature of the diagonal lengths for the sequence of the first 5 squares arranged along a gnomon. Apply Pythagorean theorem, reduction of square roots methods, and the formula for the diagonal of any width perfect square. Results are (as noted by "Cylon" the Pythagorean cult exile) for the "unit square" with sides equal to 1 each, the diagonal is the √2. Thus, according to the formula for the diagonal of perfect squares, each further iteration is the width of one side of the square multiplied by √2. According to application of the Pythagorean theorem, we arrive at √(2^2 + 2^2) = √8 (where 2^2 = 4), which reduces to the same amount expressed as 2√2. Likewise, for 3√2 = √18, of 4√2 = √32, and of 5√2 = √50. This is due to the Pythagorean theorem (a^2 + b^2 = c^2, or √ leg squared plus leg squared = √ hypotenuse), wherein √(5^2 [=25] + 5^5) = √50.