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March 2010 Difference of two square roots Is every positive integer power of consecutive integers? 2 1 2 2 1 9 8 3 50 49 2 1 the difference between the square roots of Solution This can be proved by induction, taking even powers and odd powers separately. Even powers. 2 2 1 9 8 Assume Then 2 1 2 1 2k 2k 2 [ or the starting point could be 0 2 1 1 0 ] A B 2 A2 2 B 2 where A2 2 B 2 1 2 1 2 1 3 2 2 A B 2 2 2k 3 A 4 B 2 2 A 3B 3 A 4B 2 2 2 A 3B 2 Now 3 A 4 B 2 2 A 3B A2 2 B 2 1 2 So 2 1 2 2k takes the desired form. Odd powers. 1 2 1 2 1 Assume Then 2 1 2 1 2 k 1 2 k 1 C 2 D 2C 2 D 2 where 2C 2 D 2 1 2 1 2 1 3 2 2 C 2 D 2 k 1 2 2 3C 2 D 4C 3D Now 2 3C 2 D 4C 3D 2C 2 D 2 1 2 So 2 1 2 2 k 1 also takes the desired form.