TO INFINITY AND BEYOND

A Diffy Squares Math Concepts: Subtraction, estimation, patterns, strategy. How it Works:     Draw a square, and pick four numbers to go in each of the corners. Put a dot on the midpoint of each side, and find the positive difference between the numbers on the closest corner.  Now connect the midpoints. Lo and behold, you’ve got yourself another square! Which means you can repeat the process until you get all zeroes.  The above shown is an example of a diffy square with four positive numbers 3, 1, 9 & 12.

         INFINITE SEQUENCE OF SQUARE ROOT        INFINITE SEQUENCE :                   An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integer s {1, 2, 3, ...}. Examples of infinite sequences are  N  = (0, 1, 2, 3, ...) and  S  = (1, 1/2, 1/4, 1/8, ..., 1/2  n  , ...).  INFINITE SEQUENCE OF SQUARE ROOT: Consider,                  y = x + n                (x +n)² = n² + x(x+2n)               (x + n) = √[ n² +x(x+2n)]               (x + 2n) = √[ n² +x(x+2n)(x+3n)]               ( x + 3n) = √[ n² +x(x+2n)(x+3n)(x+4n)]               ...               ...               ...               ...               ...                x+n = √[ n² +x √ n²+(n+x) √ n² +(n+2x).... ]   Remarkably the answer is exactly 3.    We have verified the pattern holds up to 4, with the next term under the radical being 6 = √6 2 . Let’s justify why the pattern will continue. Assume the pattern holds where  n  – 2 is the coef

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